Read and answer (fill in the blanks) for the questions contained in (PART 1, PART 2, & PART 4) within the attached

file:”CH6_Learning_Module_Trig_Identities”. Please answer the questions within the blanks of the attached file (CH6_Learning_Module_Trig_Identities)and upload the same file when completed. DO NOT do PART 3 and PART 5 (solving exercise problems)which have already been completed by me and are not included in your assignment. I have also uploaded the Textbook PDF files for Chapter 6 sections 1,2,& 3 to be able to answer the worksheet questions in PART 1,2,& 4.

PART 1: Contains 3 questions (attached worksheet)
PART 2: Contains 3 questions (attached worksheet)
PART 4: Contains 6 questions (attached worksheet)

Turn-In Assignment Instructions for Trig Identities Learning Module
Objective: To develop sufficient skills so that you can establish or reject whether a given trigonometric equation is an identity or not, using the textbook-listed identities in Sections 6.1 (page 606) and 6.3 (page 634).
Due: End of Module 5. Deliverable: A Nicely handwritten, scanned PDF file or a Word document using the Math Equation Writer built-into Word, Or MathType, or Google Docs using its Math Equation writer, or a Mac equivalent.
Point Value: This assignment will be worth the same as a MyMathLab Module HW.
Expected Time toComplete this Learning Module: 5 hours
So, why create the Trig Identity Learning Module and not use the MyMathLab problems available for Sections 6.1 – 6.3? MyMathLab really can’t ask the right questions, since verifying a trig identity is more of an essay than finding a specific solution. This happens to be ideal, though, for turn-in exercises. In fact, most of the learning module directs you to specific parts of the textbook to answer some questions on the reading and accomplish some basic verifications. Then, you’ll accomplish “homework problems” chosen from the textbook exercises.

Part 1: Understanding why Identities in Trigonometry are important.
Task 1: Read pages 606 – 614, hopping over the examples at this point; answer the following questions, and cite the page you found your answer:
1. What does the text say is a key purpose of using trig identities?

2. In your own words, what does the book say about solving conditional equations vs verifying identities?

3. When attempting to verify that a given equation is an identity, am I allowed to multiply or add the same quantity to both sides of the equation? Explain your reason(s).

Part 2: Identifying the most useful techniques when verifying trig identities
1. Now, review the eight examples in Section 6.1 and write down the various techniques Blitzer uses to verify that the given trig equations are identities.

2. Of the techniques you identified, which did Blitzer say was the one or two most often used? In which examples did Blitzer make use of that or those technique(s)?

3. Did any of the Fundamental Trig identities in this section define any way to change one angle into a different one?

Now, turn to page 634 (Chapter 6, Section 3) and the table of Principal Trig Identities. What is “principally” being manipulated in this set of formulas?

It is clear that the Fundamental Set of identities in Section 6.1 combined with the Principal Set of identities in Section 6.3 have a lot of math power to find EQUIVALENT Trig expressions (remember that in week one, we pointed out that Equivalent Algebraic Expressions produced equations that were Identities. We can also say the reverse: that Identities identify equivalent algebraic or trig expressions that can be substituted one for the other to solve a conditional equation.
Well, that’s what Section 6.5 is all about—Solving trig equations. SO why do we skip Section 6.4? Time, and yet another set of identities would be a bit of overkill; they can be just as useful as the others, though. And THAT’S the Big Picture for this chapter.
Part 3: Verifying some trig identities to gain familiarity with the techniques in Part 2.
These problems are to be turned in. As you work each problem, identify which of the eight examples in the section is most helpful in solving the problem, AND ALSO WHICH FUNDAMENTAL IDENTITIES YOU ARE USING TO VERIFY EACH IDENTITY in the exercises.
Section 6.1, Problems 3, 7, 11, 16, 21, 26, 31, 36, 41.
Part 4: The Most Important Trig Identity in the Universe.
From the two special triangles (the isosceles right and the 30-60-90 triangle), we know EXACT trig values for all angles that are multiples of ?/6 and ?/4 (all multiples of 30 degrees and 45 degrees, respectively). Can we find EXACT trig values for all angles that are multiples of ?/12 (15 degrees)? With the Most Important Trig Identity in the Universe, we can. In fact, all the sum and difference of angles identities, all the double angle identities and all the half-angle identities and all the Power Reduction identities begin with this Most Important Identity in the Universe.
Blitzer takes two pages to verify it.Let’s turn to page 617, the section titled “The Cosine of the Difference of Two Angles”. Much of matrix theory and vector arithmetic also depend on this formula, which is an identity.
1. First, look at Figure 1a and 1b. Describe in words what Blitzer did between 6.1a and 6.1b.

2. Why is the Unit Circle, as opposed to a larger or smaller circle centered elsewhere, preferred?

3. On page 654, Blitzer has TWO separate formulas for the length of PQ. How did he build them?

4. What Fundamental Identity plays a key role in simplifying each formula for the length of PQ?

5. What are the final steps Blitzer takes to derive the Most Important Trig Identity in the Universe?

6. State that identity here. Use a large font and Bold weight to write it.

Part 5:Practice using it and the related sum/difference identities for sine and tangent. Same rules apply as for Part 3.
Exercise 6.2, Problems 2, 3, 9, 15, 24, 30, 33, 36, 39 (pp. 623-624). The last three are VERY IMPORTANT identities.Add in three problems from 57 – 63of your choosing(p. 624), but find only “Part a” for one of them, “Part b” for the second of them, and “Part c” for the third of them. QUADRANT INFO IS VERY IMPORTANT HERE.

That completes the assignment. Upload your electronic file containing the answers to all the Concept Questions and the Exercises from Parts 3 and 5 (some, but not many, of the Section 6.3 exercises also happen to be problems in the MyMathLab Module 5 HW. Hey, you get two shots at those problems.) If for some reason, your upload fails, send it as an attachment via email to your instructor. YOUR FILE NAME SHOULD BE LASTNAME LEARNING MODULE.extension like .docx, .doc, .pdf. MAC USERS: .doc or pdf, please. Your instructor may or may not have a Mac.

Analytic Trigonometry

You enjoy watching your friend participate in the shot put at college track and field events. After a few full turns in a circle, he throws (“puts”) an 8-pound, 13-ounce shot from the shoulder. The range of his throwing distance continues to improve. Knowing that you are studying trigonometry, he asks if there is some way that a trigonometric expression might help achieve his best possible distance in the event.

6
605

This problem appears as Exercise 79 in Exercise Set 6.3. In the solution, you will obtain critical information about athletic performance using a trigonometric identity. In this chapter, we derive important categories of identities involving trigonometric functions. You will learn how to use these identities to better understand your periodic world.

606 Chapter 6 Analytic Trigonometry Section Objective ? Use the fundamental
trigonometric identities to verify identities.

6.1 Verifying Trigonometric Identities
o you enjoy solving puzzles? The process is a natural way to develop problem-solving skills that are important in every area of our lives. Engaging in problem solving for sheer pleasure releases chemicals in the brain that enhance our feeling of well-being. Perhaps this is why puzzles have fascinated people for over 12,000 years. Thousands of relationships exist among the six trigonometric functions. Verifying these relationships is like solving a puzzle. Why? There are no rigid rules for the process. Thus, proving a trigonometric relationship requires you to be creative in your approach to problem solving. By learning to establish these relationships, you will become a better, more confident problem solver. Furthermore, you may enjoy the feeling of satisfaction that accompanies solving each “puzzle.”

The Fundamental Identities
In Chapter 5, we used right triangles to establish relationships among the trigonometric functions. Although we limited domains to acute angles, the fundamental identities listed in the following box are true for all values of x for which the expressions are defined.

Fundamental Trigonometric Identities
Reciprocal Identities 1 csc x 1 sin x 1 sec x 1 cos x 1 cot x 1 cot x = tan x tan x = cos x sin x

Study Tip
Memorize the identities in the box. You may need to use variations of these fundamental identities. For example, instead of sin2 x + cos2 x = 1 you might want to use sin2 x = 1 – cos2 x or cos2 x = 1 – sin2 x. Therefore, it is important to know each relationship well so that mental algebraic manipulation is possible.

sin x = csc x = Quotient Identities

cos x = sec x =

tan x = Pythagorean Identities

sin x cos x

cot x =

sin2 x + cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = csc2 x Even-Odd Identities sin1 – x2 = – sin x cos1 – x2 = cos x sec1 – x2 = sec x tan1 – x2 = – tan x cot1 – x2 = – cot x

csc1 – x2 = – csc x

Use the fundamental trigonometric identities to verify identities.

Using Fundamental Identities to Verify Other Identities
The fundamental trigonometric identities are used to establish other relationships among trigonometric functions. To verify an identity, we show that one side of the identity can be simplified so that it is identical to the other side. Each side of the equation is manipulated independently of the other side of the equation. Start with the side containing the more complicated expression. If you substitute one or more

Section 6.1 Verifying Trigonometric Identities

607

fundamental identities on the more complicated side, you will often be able to rewrite it in a form identical to that of the other side. No one method or technique can be used to verify every identity. Some identities can be verified by rewriting the more complicated side so that it contains only sines and cosines.

EXAMPLE 1
Verify the identity:

Changing to Sines and Cosines to Verify an Identity
sec x cot x = csc x.

Solution The left side of the equation contains the more complicated expression. Thus, we work with the left side. Let us express this side of the identity in terms of sines and cosines. Perhaps this strategy will enable us to transform the left side into csc x, the expression on the right.
1 # cos x sec x cot x = cos x sin x
1

1 Apply a reciprocal identity: sec x = and a cos x cos x quotient identity: cot x = . sin x Divide both the numerator and the denominator by cos x, the common factor. Multiply the remaining factors in the numerator and denominator. 1 Apply a reciprocal identity: csc x = . sin x

1 = cos x
1

cos x sin x

1 = sin x = csc x

By working with the left side and simplifying it so that it is identical to the right side, we have verified the given identity.

Technology
Numeric and Graphic Connections You can use a graphing utility to provide evidence of an identity. Enter each side of the identity separately under y1 and y2 . Then use the ?TABLE ? feature or the graphs. The table should show that the function values are the same except for those values of x for which y1 , y2 , or both, are undefined. The graphs should appear to be identical. Let’s check the identity in Example 1:

sec x cot x=csc x.
y1 = sec x cot x Enter sec x as 1 cos x and cot x as 1 . tan x y2 = csc x Enter csc x as 1 . sin x

Numeric Check Display a table for y1 and y2 . We started p our table at – p and used ¢ Tbl = . 8
y1 = sec x cot x y2 = csc x

Graphic Check Display graphs for y1 and y2 .
y1 = sec x cot x and y2 = csc x
1
-p 2 p 2

-1

[-p, p, q] by [-4, 4, 1] The graphs appear to be identical. Function values are the same except for values of x for which y1, y2, or both, are undefined.

608 Chapter 6 Analytic Trigonometry

Check Point

Verify the identity:

csc x tan x = sec x.

In verifying an identity, stay focused on your goal. When manipulating one side of the equation, continue to look at the other side to keep the desired form of the result in mind.

Study Tip
Verifying that an equation is an identity is different from solving an equation. You do not verify an identity by adding, subtracting, multiplying, or dividing each side by the same expression. If you do this, you have already assumed that the given statement is true. You do not know that it is true until after you have verified it.

EXAMPLE 2
Verify the identity:

Changing to Sines and Cosines to Verify an Identity
sin x tan x + cos x = sec x.

Solution The left side is more complicated, so we start with it. Notice that the left side contains the sum of two terms, but the right side contains only one term. This means that somewhere during the verification process, the two terms on the left side must be added to form one term. Let’s begin by expressing the left side of the identity so that it contains sin x only sines and cosines. Thus, we apply a quotient identity and replace tan x by . cos x Perhaps this strategy will enable us to transform the left side into sec x, the expression on the right.

Study Tip
When proving identities, be sure to write the variable associated with each trigonometric function. Do not get lazy and write sin tan + cos for sin x tan x + cos x because sin, tan, and cos are meaningless without specified variables.

sin x tan x + cos x = sin x a =

sin x b + cos x cos x

Apply a quotient identity: tan x = Multiply.

sin x . cos x

sin2 x + cos x cos x

sin2 x cos x = + cos x # cos x cos x = sin2 x cos2 x + cos x cos x sin2 x + cos2 x cos x 1 cos x

The least common denominator is cos x. Write the second expression with a denominator of cos x. Multiply. Add numerators and place this sum over the least common denominator. Apply a Pythagorean identity: sin2 x + cos2 x = 1. Apply a reciprocal identity: sec x = 1 . cos x

= sec x

By working with the left side and arriving at the right side, the identity is verified.

Check Point

Verify the identity:

cos x cot x + sin x = csc x.

Some identities are verified using factoring to simplify a trigonometric expression.

EXAMPLE 3
Verify the identity:

Using Factoring to Verify an Identity
cos x – cos x sin2 x = cos3 x.

Section 6.1 Verifying Trigonometric Identities

609

Solution We start with the more complicated side, the left side. Factor out the greatest common factor, cos x, from each of the two terms.
cos x – cos x sin2 x = cos x11 – sin2 x2 = cos x # cos2 x
Factor cos x from the two terms. Use a variation of sin2 x + cos2 x = 1. Solving for cos2 x, we obtain cos2 x = 1 – sin2 x. Multiply.

= cos3 x

We worked with the left side and arrived at the right side. Thus, the identity is verified.

Check Point

Verify the identity:

sin x – sin x cos2 x = sin3 x.

There is often more than one technique that can be used to verify an identity.

EXAMPLE 4
Verify the identity:

Using Two Techniques to Verify an Identity
1 + sin u = sec u + tan u. cos u

Solution
Method 1. Separating a Single-Term Quotient into Two Terms Let’s separate the quotient on the left side into two terms using a b a + b = + . c c c Perhaps this strategy will enable us to transform the left side into sec u + tan u, the sum on the right. 1 + sin u 1 sin u = + cos u cos u cos u = sec u + tan u
Divide each term in the numerator by cos u. Apply a reciprocal identity and a quotient identity: 1 sin u sec u = and tan u = . cos u cos u

We worked with the left side and arrived at the right side.Thus, the identity is verified. Method 2. Changing to Sines and Cosines Let’s work with the right side of the identity and express it so that it contains only sines and cosines. sec u + tan u = 1 sin u + cos u cos u
1 + sin u cos u Apply a reciprocal identity and a quotient identity: 1 sin u sec u = and tan u = . cos u cos u Add numerators. Put this sum over the common denominator.

We worked with the right side and arrived at the left side. Thus, the identity is verified.

Check Point

Verify the identity:

1 + cos u = csc u + cot u. sin u

How do we verify identities in which sums or differences of fractions with trigonometric functions appear on one side? Use the least common denominator and combine the fractions. This technique is especially useful when the other side of the identity contains only one term.

610 Chapter 6 Analytic Trigonometry EXAMPLE 5
Verify the identity:

Combining Fractional Expressions to Verify an Identity
cos x 1 + sin x + = 2 sec x. cos x 1 + sin x

Solution We start with the more complicated side, the left side. The least common

denominator of the fractions is 11 + sin x21cos x2. We express each fraction in terms of this least common denominator by multiplying the numerator and denominator by the extra factor needed to form 11 + sin x21cos x2. cos x 1 + sin x + cos x 1 + sin x = 11 + sin x21cos x2 cos x1cos x2 + 11 + sin x211 + sin x2 11 + sin x21cos x2
The least common denominator is 11 + sin x21cos x2. Rewrite each fraction with the least common denominator. Use the FOIL method to multiply 11 + sin x211 + sin x2. Add numerators. Put this sum over the least common denominator. Regroup terms to apply a Pythagorean identity. Apply a Pythagorean identity: sin2 x + cos2 x = 1. Add constant terms in the numerator: 1 + 1 = 2. Factor and simplify.

Study Tip
Some students have difficulty verifying identities due to problems working with fractions. If this applies to you, review the section on rational expressions in Chapter P.

cos2 x 1 + 2 sin x + sin2 x + 11 + sin x21cos x2 11 + sin x21cos x2 cos2 x + 1 + 2 sin x + sin2 x 11 + sin x21cos x2 11 + sin x21cos x2

1sin2 x + cos2 x2 + 1 + 2 sin x 1 + 1 + 2 sin x 11 + sin x21cos x2 2 + 2 sin x 11 + sin x21cos x2 11 + sin x2 1cos x2 2 cos x 2 11 + sin x2

= 2 sec x

Apply a reciprocal identity: 1 sec x = . cos x

We worked with the left side and arrived at the right side. Thus, the identity is verified.

Check Point

Verify the identity:

sin x 1 + cos x + = 2 csc x. 1 + cos x sin x

Some identities are verified using a technique that may remind you of rationalizing a denominator.

EXAMPLE 6

Multiplying the Numerator and Denominator by the Same Factor to Verify an Identity
1 – cos x sin x = . 1 + cos x sin x

Verify the identity:

Solution The suggestions given in the previous examples do not apply here. Everything is already expressed in terms of sines and cosines. Furthermore, there are no fractions to combine and neither side looks more complicated than the other. Let’s solve the puzzle by working with the left side and making it look like the

Section 6.1 Verifying Trigonometric Identities

611

expression on the right. The expression on the right contains 1 – cos x in the numerator. This suggests multiplying the numerator and denominator of the left side by 1 – cos x. By doing this, we obtain a factor of 1 – cos x in the numerator, as in the numerator on the right.

Discovery
Verify the identity in Example 6 by making the right side look like the left side. Start with the expression on the right. Multiply the numerator and denominator by 1 + cos x.

sin x # 1 – cos x sin x = Multiply numerator and denominator by 1 – cos x. 1 + cos x 1 + cos x 1 – cos x = sin x11 – cos x2 sin x11 – cos x2 sin x 1 – cos x = sin x
2

1 – cos x

Multiply. Use 1A + B21A – B2 = A2 – B2, with A = 1 and B = cos x, to multiply denominators. Use a variation of sin2 x + cos2 x = 1. Solving for sin2 x, we obtain sin2 x = 1 – cos2 x. Simplify: sin x sin x 1 = = . 2 # sin x sin x sin x sin x

We worked with the left side and arrived at the right side. Thus, the identity is verified.

Check Point
EXAMPLE 7

Verify the identity:

cos x 1 – sin x . = cos x 1 + sin x

Changing to Sines and Cosines to Verify an Identity
tan x – sin1 – x2 1 + cos x = tan x.

Verify the identity:

Solution We begin with the left side. Our goal is to obtain tan x, the expression
on the right.

Discovery
Try simplifying sin x + sin x cos x 1 + cos x by multiplying the two terms in the numerator and the two terms in the denominator by cos x. This method for simplifying the complex fraction involves multiplying the numerator and the denominator by the least common denominator of all fractions in the expression. Do you prefer this simplification procedure over the method used on the right?

tan x – sin1 – x2 1 + cos x

tan x – 1 – sin x2 1 + cos x

The sine function is odd: sin1 – x2 = – sin x.

tan x + sin x 1 + cos x sin x + sin x cos x = 1 + cos x sin x sin x cos x + cos x cos x = 1 + cos x sin x + sin x cos x cos x = 1 + cos x sin x + sin x cos x 1 + cos x = , cos x 1 sin x + sin x cos x # 1 = cos x 1 + cos x = = sin x 11 + cos x2
1

Simplify. Apply a quotient identity: sin x tan x = . cos x Express the terms in the numerator with the least common denominator, cos x.

Add in the numerator. Rewrite the main fraction bar as , . Invert the divisor and multiply.

cos x

1 1 + 1 cos x

Factor and simplify. Multiply the remaining factors in the numerator and in the denominator. Apply a quotient identity.

sin x cos x

= tan x

The left side simplifies to tan x, the right side. Thus, the identity is verified.

612 Chapter 6 Analytic Trigonometry

Check Point

Verify the identity:

sec x + csc1 – x2 sec x csc x

= sin x – cos x.

Is every identity verified by working with only one side? No. You can sometimes work with each side separately and show that both sides are equal to the same trigonometric expression. This is illustrated in Example 8.

EXAMPLE 8

Working with Both Sides Separately to Verify an Identity
1 1 + = 2 + 2 cot2 u. 1 + cos u 1 – cos u
The least common denominator is 11 + cos u211 – cos u2. Rewrite each fraction with the least common denominator. Add numerators. Put this sum over the least common denominator. Simplify the numerator: – cos u + cos u = 0 and 1 + 1 = 2. Multiply the factors in the denominator.

Verify the identity:

Solution We begin by working with the left side.
1 1 + 1 + cos u 1 – cos u = 11 + cos u211 – cos u2 1 – cos u + 1 + cos u 11 + cos u211 – cos u2 2 11 + cos u211 – cos u2 2 1 – cos2 u 111 – cos u2 + 11 + cos u211 – cos u2 111 + cos u2

Now we work with the right side. Our goal is to transform this side into the simplified 2 form attained for the left side, . 1 – cos2 u 2 + 2 cot2 u = 2 + 2 ¢ cos2 u = sin2 u
Use a quotient identity: cot u = cos u . sin u

2 sin2 u 2 cos2 u + sin2 u sin2 u 2 sin2 u + 2 cos2 u sin2 u 21sin2 u + cos2 u2 sin2 u 2 sin2 u 2 1 – cos2 u

Rewrite each term with the least common denominator, sin2 u. Add numerators. Put this sum over the least common denominator. Factor out the greatest common factor, 2. Apply a Pythagorean identity: sin2 u + cos2 u = 1. Use a variation of sin2 u + cos2 u = 1 and solve for sin2 u: sin2 u = 1 – cos2 u.

The identity is verified because both sides are equal to

2 . 1 – cos2 u

Check Point

8 Verify the identity:

1 1 + = 2 + 2 tan2 u. 1 + sin u 1 – sin u

Section 6.1 Verifying Trigonometric Identities

613

Guidelines for Verifying Trigonometric Identities
There is often more than one correct way to solve a puzzle, although one method may be shorter and more efficient than another. The same is true for verifying an identity. For example, how would you verify csc2 x – 1 = cos2 x? csc2 x One approach is to use a Pythagorean identity, 1 + cot2 x = csc2 x, on the left side. Then change the resulting expression to sines and cosines. cos2 x sin2 x cot2 x cos2 x sin2 x csc2 x-1 (1+cot2 x)-1 = = = 2 ? =cos2 x 2 2 2 = 1 csc x csc x csc x sin x 1 sin2 x Apply a Pythagorean identity:
1 + cot2 x = csc2 x. Use cot x = cos x and sin x csc x = 1 to change sin x to sines and cosines. Invert the divisor and multiply.

A more efficient strategy for verifying this identity may not be apparent at first glance. Work with the left side and divide each term in the numerator by the denominator, csc2 x. csc2 x csc2 x-1 1 = =1-sin2 x=cos2 x 2 csc x csc2 x csc2 x
identity: sin x = Apply a reciprocal 1 . csc x Use sin2 x + cos2 x = 1 and solve for cos2 x.

With this strategy, we again obtain cos2 x, the expression on the right side, and it takes fewer steps than the first approach. An even longer strategy, but one that works, is to replace each of the two 1 occurrences of csc2 x on the left side by 2 . This may be the approach that you first sin x consider, particularly if you become accustomed to rewriting the more complicated side in terms of sines and cosines. The selection of an appropriate fundamental identity to solve the puzzle most efficiently is learned through lots of practice. The more identities you prove, the more confident and efficient you will become. Although practice is the only way to learn how to verify identities, there are some guidelines developed throughout the section that should help you get started.

Guidelines for Verifying Trigonometric Identities
• Work with each side of the equation independently of the other side. Start with the more complicated side and transform it in a step-by-step fashion until it looks exactly like the other side. • Analyze the identity and look for opportunities to apply the fundamental identities. • Try using one or more of the following techniques: 1. Rewrite the more complicated side in terms of sines and cosines. 2. Factor out the greatest common factor. 3. Separate a single-term quotient into two terms: a b a – b a b a + b = + and = – . c c c c c c 4. Combine fractional expressions using the least common denominator. 5. Multiply the numerator and the denominator by a binomial factor that appears on the other side of the identity. • Don’t be afraid to stop and start over again if you are not getting anywhere. Creative puzzle solvers know that strategies leading to dead ends often provide good problem-solving ideas.

614 Chapter 6 Analytic Trigonometry

Exercise Set 6.1
Practice Exercises
In Exercises 1–60, verify each identity. 1. sin x sec x = tan x 3. tan1 – x2 cos x = – sin x 5. tan x csc x cos x = 1 7. sec x – sec x sin x = cos x 8. csc x – csc x cos2 x = sin x 9. cos x – sin x = 1 – 2 sin x 10. cos2 x – sin2 x = 2 cos2 x – 1 11. csc u – sin u = cot u cos u 12. tan u + cot u = sec u csc u 13. tan u cot u = sin u csc u 14. cos u sec u = tan u cot u
2 2 2 2

40. cot2 2x + cos2 2x + sin2 2x = csc2 2x 2. cos x csc x = cot x 4. cot1 – x2 sin x = – cos x 6. cot x sec x sin x = 1 44. 41. 43. tan 2u + cot 2u = sec 2u csc 2u 42. tan 2u + cot 2u = csc 2u sec 2u

tan x + tan y sin x cos y + cos x sin y = 1 – tan x tan y cos x cos y – sin x sin y cot x + cot y cos x sin y + sin x cos y = 1 – cot x cot y sin x sin y – cos x cos y 1 – sin x 1 + sin x 1 – cos x 1 + cos x 48. cot t csc t – 1 = cot t csc t + 1

45. 1sec x – tan x22 = 46. 1csc x – cot x22 = 47. 49. 50.

sec t + 1 tan t = tan t sec t – 1 1 + cos t = 1csc t + cot t22 1 – cos t

15. sin2 u11 + cot2 u2 = 1 17. sin t tan t =
2

16. cos2 u11 + tan2 u2 = 1 18. cos t cot t =
2

1 – cos t cos t

1 – sin t sin t

cos2 t + 4 cos t + 4 2 sec t + 1 = cos t + 2 sec t

51. cos4 t – sin4 t = 1 – 2 sin2 t 52. sin4 t – cos4 t = 1 – 2 cos2 t 53. 54. sin u – cos u cos u – sin u + = 2 – sec u csc u sin u cos u sin u cos u = sin u + cos u 1 – cot u tan u – 1

19. 21.

csc t = csc t sec t cot t tan2 t = sec t – cos t sec t

20. 22.

sec t = sec t csc t tan t cot2 t = csc t – sin t csc t

1 – cos u = csc u – cot u 23. sin u sin t cos t + = 1 25. csc t sec t 26. sin t cos t + = sin t + cos t tan t cot t

1 – sin u = sec u – tan u 24. cos u

55. 1tan2 u + 121cos2 u + 12 = tan2 u + 2 56. 1cot2 u + 121sin2 u + 12 = cot2 u + 2 57. 1cos u – sin u22 + 1cos u + sin u22 = 2

cos t = sec t 27. tan t + 1 + sin t 29. 1 31. 32. sin2 x = cos x 1 + cos x

sin t = csc t 28. cot t + 1 + cos t 30. 1 cos2 x = sin x 1 + sin x

58. 13 cos u – 4 sin u22 + 14 cos u + 3 sin u22 = 25 59. 60. cos2 x – sin2 x = cos2 x 1 – tan2 x cos x – sin x sin x + cos x = sec x csc x cos x sin x

cos x 1 – sin x = 2 sec x + cos x 1 – sin x sin x cos x – 1 + = 0 cos x + 1 sin x

Practice Plus
In Exercises 61–66, half of an identity and the graph of this half are given. Use the graph to make a conjecture as to what the right side of the identity should be. Then prove your conjecture. 61. 1sec x + tan x21sec x – tan x2 sec x = ?

33. sec2 x csc2 x = sec2 x + csc2 x 34. csc2 x sec x = sec x + csc x cot x 35. 36. 37. 38. sec x – csc x tan x – 1 = sec x + csc x tan x + 1 csc x – sec x cot x – 1 = csc x + sec x cot x + 1 sin2 x – cos2 x = sin x – cos x sin x + cos x tan2 x – cot2 x = tan x – cot x tan x + cot x

39. tan2 2x + sin2 2x + cos2 2x = sec2 2x

[-2p, 2p, q] by [-4, 4, 1]

Section 6.1 Verifying Trigonometric Identities
62. sec2 x csc x = ? sec2 x + csc2 x

615

In Exercises 67–74, rewrite each expression in terms of the given function or functions. 67. 69. 71. tan x + cot x ; cos x csc x cos x + tan x; cos x 1 + sin x 1 cos x ; csc x 1 – cos x 1 + cos x 68. 70. sec x + csc x ; sin x 1 + tan x 1 – cot x; cot x sin x cos x

[-2p, 2p, q] by [-4, 4, 1]

72. 1sec x + csc x21sin x + cos x2 – 2 – cot x; tan x 73. 74. 1 ; sec x and tan x csc x – sin x 1 – sin x 1 + sin x ; sec x and tan x 1 + sin x 1 – sin x

cos x + cot x sin x 63. = ? cot x

Writing in Mathematics
75. Explain how to verify an identity. 76. Describe two strategies that can be used to verify identities.
[-2p, 2p, q] by [-4, 4, 1]

77. Describe how you feel when you successfully verify a difficult identity. What other activities do you engage in that evoke the same feelings? 78. A 10-point question on a quiz asks students to verify the identity sin2 x – cos2 x = sin x – cos x. sin x + cos x One student begins with the left side and obtains the right side as follows: sin2 x – cos2 x sin2 x cos2 x = = sin x – cos x. cos x sin x + cos x sin x

64.

cos x tan x – tan x + 2 cos x – 2 = ? tan x + 2

[-2p, 2p, q] by [-4, 4, 1]

How many points (out of 10) would you give this student? Explain your answer.

65.

1 1 + = ? sec x + tan x sec x – tan x

Technology Exercises
In Exercises 79–87, graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of x for which both sides are defined but not equal.

[-2p, 2p, q] by [-4, 4, 1]

79. tan x = sec x1sin x – cos x2 + 1 80. sin x = – cos x tan1 – x2 81. sin a x + 82. cos a x + p p b = sin x + sin 4 4 p p b = cos x + cos 4 4 84. sin1x + p2 = sin x

66.

1 + cos x sin x + = ? sin x 1 + cos x

83. cos1x + p2 = cos x 85. sin x = csc x 1 – cos2 x

86. sin x – sin x cos2 x = sin3 x
[-2p, 2p, q] by [-4, 4, 1]

87. 3sin2 x + cos2 x = sin x + cos x

616 Chapter 6 Analytic Trigonometry

Critical Thinking Exercises
Make Sense? In Exercises 88–91, determine whether each
statement makes sense or does not make sense, and explain your reasoning. 88. The word identity is used in different ways in additive identity, multiplicative identity, and trigonometric identity. 89. To prove a trigonometric identity, I select one side of the equation and transform it until it is the other side of the equation, or I manipulate both sides to a common trigonometric expression. 90. In order to simplify cos x sin x , I need to know how cos x 1 – sin x to subtract rational expressions with unlike denominators. way that I can simplify

Group Exercise
97. Group members are to write a helpful list of items for a pamphlet called “The Underground Guide to Verifying Identities.” The pamphlet will be used primarily by students who sit, stare, and freak out every time they are asked to verify an identity. List easy ways to remember the fundamental identities. What helpful guidelines can you offer from the perspective of a student that you probably won’t find in math books? If you have your own strategies that work particularly well, include them in the pamphlet.

Preview Exercises
Exercises 98–100 will help you prepare for the material covered in the next section. 98. Give exact values for cos 30°, sin 30°, cos 60°, sin 60°, cos 90°, and sin 90°. 99. Use the appropriate values from Exercise 98 to answer each of the following. a. Is cos 130° + 60°2, or cos 90°, equal to cos 30° + cos 60°? b. Is cos 130° + 60°2, or cos 90°, equal to cos 30° cos 60° – sin 30° sin 60°?

91. The most efficient 1sec x + 121sec x – 12

is to immediately rewrite the sin2 x expression in terms of cosines and sines.

In Exercises 92–95, verify each identity. sin3 x – cos3 x 92. = 1 + sin x cos x sin x – cos x 93. sin x – cos x + 1 sin x + 1 = cos x sin x + cos x – 1 95. ln etan
2

100. Use the appropriate values from Exercise 98 to answer each of the following.
x – sec2 x

94. ln ƒ sec x ƒ = – ln ƒ cos x ƒ

= -1

a. Is sin 130° + 60°2, or sin 90°, equal to sin 30° + sin 60°? b. Is sin 130° + 60°2, or sin 90°, equal to sin 30° cos 60° + cos 30° sin 60°?

96. Use one of the fundamental identities in the box on page 606 to create an original identity.

Section Objectives ? Use the formula for the ? ?

6.2

Sum and Difference Formulas
isten to the same note played on a piano and a violin. The notes have a different quality or “tone.” Tone depends on the way an instrument vibrates. However, the less than 1% of the population with amusia, or true tone deafness, cannot tell the two sounds apart. Even simple, familiar tunes such as Happy Birthday and Jingle Bells are mystifying to amusics. When a note is played, it vibrates at a specific fundamental frequency and has a particular amplitude. Amusics cannot tell the difference between two sounds from tuning forks modeled by p = 3 sin 2t and p = 2 sin12t + p2, respectively. However, they can recognize the difference between the two equations. Notice that the second equation contains the sine of the sum of two angles. In this section, we will be developing identities involving the sums or differences of two angles. These formulas are called the sum and difference formulas. We begin with cos1a – b 2, the cosine of the difference of two angles.

cosine of the difference of two angles. Use sum and difference formulas for cosines and sines. Use sum and difference formulas for tangents.

616 Chapter 6 Analytic Trigonometry

91. The most efficient 1sec x + 121sec x – 12

is to immediately rewrite the sin2 x expression in terms of cosines and sines.

In Exercises 92–95, verify each identity. sin3 x – cos3 x 92. = 1 + sin x cos x sin x – cos x 93. sin x – cos x + 1 sin x + 1 = cos x sin x + cos x – 1 95. ln etan
2

100. Use the appropriate values from Exercise 98 to answer each of the following.
x – sec2 x

94. ln ƒ sec x ƒ = – ln ƒ cos x ƒ

= -1

a. Is sin 130° + 60°2, or sin 90°, equal to sin 30° + sin 60°? b. Is sin 130° + 60°2, or sin 90°, equal to sin 30° cos 60° + cos 30° sin 60°?

96. Use one of the fundamental identities in the box on page 606 to create an original identity.

Section Objectives ? Use the formula for the ? ?

6.2

cosine of the difference of two angles. Use sum and difference formulas for cosines and sines. Use sum and difference formulas for tangents.

Section 6.2 Sum and Difference Formulas
y Q = (cos a, sin a) P = (cos ß, sin ß)

617

The Cosine of the Difference of Two Angles
The Cosine of the Difference of Two Angles
x

a a-b

cos1a – b 2 = cos a cos b + sin a sin b The cosine of the difference of two angles equals the cosine of the first angle times the cosine of the second angle plus the sine of the first angle times the sine of the second angle. We use Figure 6.1 to prove the identity in the box. The graph in Figure 6.1(a) shows a unit circle, x2 + y2 = 1. The figure uses the definitions of the cosine and sine functions as the x- and y-coordinates of points along the unit circle. For example, point P corresponds to angle b . By definition, the x-coordinate of P is cos b and the y-coordinate is sin b . Similarly, point Q corresponds to angle a. By definition, the x-coordinate of Q is cos a and the y-coordinate is sin a. Note that if we draw a line segment between points P and Q, a triangle is formed. Angle a – b is one of the angles of this triangle. What happens if we rotate this triangle so that point P falls on the x-axis at (1, 0)? The result is shown in Figure 6.1(b). This rotation changes the coordinates of points P and Q. However, it has no effect on the length of line segment PQ. We can use the distance formula, d = 41×2 – x122 + 1y2 – y122 , to find an expression for PQ in Figure 6.1(a) and in Figure 6.1(b). By equating the two expressions for PQ, we will obtain the identity for the cosine of the difference of two angles, a – b . We first apply the distance formula in Figure 6.1(a).

x2 + y2 = 1 (a) y Q = (cos (a – ß), sin (a – ß)) P = (1, 0) x a-b

x2 (b)

Figure 6.1 Using the unit circle
and PQ to develop a formula for cos1a – b 2

PQ = 41cos a – cos b 22 + 1sin a – sin b 22

Apply the distance formula,

d = 41×2 – x122 + 1y2 – y122 , to find the distance between 1cos b , sin b 2 and 1cos a, sin a2. Square each expression using 1A – B22 = A2 – 2AB + B2. Regroup terms to apply a Pythagorean identity. Because sin2 x + cos2 x = 1, each expression in parentheses equals 1. Simplify.

= 3cos2 a – 2 cos a cos b + cos2 b + sin2 a – 2 sin a sin b + sin2 b = 41sin2 a + cos2 a2 + 1sin2 b + cos2 b 2 – 2 cos a cos b – 2 sin a sin b = 21 + 1 – 2 cos a cos b – 2 sin a sin b = 22 – 2 cos a cos b – 2 sin a sin b

Next, we apply the distance formula in Figure 6.1(b) to obtain a second expression for PQ. We let 1×1 , y12 = 11, 02 and 1×2 , y22 = 1cos1a – b 2, sin1a – b 22. PQ = 43cos1a – b 2 – 142 + 3sin1a – b 2 – 042 = 4cos21a – b 2 – 2 cos1a – b 2 + 1 + sin21a – b 2 =?cos2 (a-b)-2 cos (a-b)+1+sin2 (a-b)
Using a Pythagorean identity, sin2 (a – b) + cos2 (a – b) = 1.

Apply the distance formula to find the distance between (1, 0) and 1cos1a – b 2, sin1a – b 22. Square each expression.

=?1-2 cos (a-b)+1 = 42 – 2 cos1a – b 2

Use a Pythagorean identity. Simplify.

618 Chapter 6 Analytic Trigonometry
Now we equate the two expressions for PQ. 42 – 2 cos1a – b 2 = 22 – 2 cos a cos b – 2 sin a sin b 2 – 2 cos1a – b 2 = 2 – 2 cos a cos b – 2 sin a sin b – 2 cos1a – b 2 = – 2 cos a cos b – 2 sin a sin b
The rotation does not change the length of PQ. Square both sides to eliminate radicals. Subtract 2 from both sides of the equation. Divide both sides of the equation by – 2.

cos1a – b 2 = cos a cos b + sin a sin b Use the formula for the cosine of the difference of two angles.

Sound Quality and Amusia
People with true tone deafness cannot hear the difference among tones produced by a tuning fork, a flute, an oboe, and a violin. They cannot dance or tell the difference between harmony and dissonance. People with amusia appear to have been born without the wiring necessary to process music. Intriguingly, they show no overt signs of brain damage and their brain scans appear normal. Thus, they can visually recognize the difference among sound waves that produce varying sound qualities. Varying Sound Qualities • Tuning fork: Sound waves are rounded and regular, giving a pure and gentle tone.

This proves the identity for the cosine of the difference of two angles. Now that we see where the identity for the cosine of the difference of two angles comes from, let’s look at some applications of this result.

EXAMPLE 1

Using the Difference Formula for Cosines to Find an Exact Value

Find the exact value of cos 15°.

Solution We know exact values for trigonometric functions of 60° and 45°. Thus, we write 15° as 60° – 45° and use the difference formula for cosines.
cos 15° = cos160° – 45°2 = cos 60° cos 45° + sin 60° sin 45° cos1a – b 2 = cos a cos b + sin a sin b = 1 # 22 23 # 22 + 2 2 2 2 22 26 + 4 4 22 + 26 4
Substitute exact values from memory or use special right triangles. Multiply.

Add.

23 . Obtain this exact value using 2 cos 30° = cos190° – 60°2 and the difference formula for cosines.

Check Point

We know that cos 30° =

• Flute: Sound waves are smooth and give a fluid tone.

EXAMPLE 2

Using the Difference Formula for Cosines to Find an Exact Value

• Oboe: Rapid wave changes give a richer tone.

Find the exact value of cos 80° cos 20° + sin 80° sin 20°.

Solution The given expression is the right side of the formula for cos1a – b 2 with a = 80° and b = 20°.
cos (a – b) = cos a cos b + sin a sin b

• Violin: Jagged waves give a brighter harsher tone.

cos 80? cos 20? +sin 80? sin 20? =cos (80? -20? )=cos 60? =

1 2

Check Point

Find the exact value of cos 70° cos 40° + sin 70° sin 40°.

Section 6.2 Sum and Difference Formulas

619

EXAMPLE 3
Verify the identity:

Verifying an Identity
cos1a – b 2 sin a cos b = cot a + tan b .

Solution We work with the left side.
cos1a – b 2 sin a cos b = cos a cos b + sin a sin b sin a cos b cos a cos b sin a sin b + sin a cos b sin a cos b cos a # cos b sin a # sin b + sin a cos b sin a cos b
Use the formula for cos1a – b 2. Divide each term in the numerator by sin a cos b . This step can be done mentally. We wanted you to see the substitutions that follow. Use quotient identities. Simplify.

= cot a # 1 + 1 # tan b = cot a + tan b

Technology
Graphic Connections The graphs of p y = cos a – x b 2 and y = sin x are shown in the same viewing rectangle. The graphs are the same. The displayed math on the right with the voice balloon on top shows the equivalence algebraically.
y = cos p 2 -x and y = sin x

We worked with the left side and arrived at the right side. Thus, the identity is verified.

Check Point

Verify the identity:

cos1a – b 2 cos a cos b

= 1 + tan a tan b .

The difference formula for cosines is used to establish other identities. For example, in our work with right triangles, we noted that cofunctions of complements p are equal. Thus, because – u and u are complements, 2 cos a p – u b = sin u. 2

(

)
1

We can use the formula for cos1a – b 2 to prove this cofunction identity for all angles.
Apply cos (a – b) with a = p 2 and b = u. cos (a – b) = cos a cos b + sin a sin b
p

-p -1

p p p cos a -ub =cos cos u+sin sin u 2 2 2 = 0 # cos u + 1 # sin u = sin u

Use sum and difference formulas for cosines and sines.

Sum and Difference Formulas for Cosines and Sines

Our formula for cos1a – b 2 can be used to verify an identity for a sum involving cosines, as well as identities for a sum and a difference for sines.

Sum and Difference Formulas for Cosines and Sines
1. 2. 3. 4. cos1a + cos1a sin1a + sin1a b 2 = cos a cos b – sin a sin b b 2 = cos a cos b + sin a sin b b 2 = sin a cos b + cos a sin b b 2 = sin a cos b – cos a sin b

620 Chapter 6 Analytic Trigonometry
Up to now, we have concentrated on the second formula in the box on the previous page, cos1a – b 2 = cos a cos b + sin a sin b . The first identity gives a formula for the cosine of the sum of two angles. It is proved as follows: cos1a + b 2 = cos3a – 1 – b 24 = cos a cos b + sin a1 – sin b 2 = cos a cos b – sin a sin b .
Express addition as subtraction of an additive inverse. Cosine is even: cos1 – b 2 = cos b . Sine is odd: sin1 – b 2 = – sin b . Simplify.

= cos a cos1 – b 2 + sin a sin1 – b 2 Use the difference formula for cosines.

Thus, the cosine of the sum of two angles equals the cosine of the first angle times the cosine of the second angle minus the sine of the first angle times the sine of the second angle. The third identity in the box gives a formula for sin 1a + b 2, the sine of the sum of two angles. It is proved as follows: sin1a + b 2 = cos c p – 1a + b 2 d 2 p – ab – b R 2
Use a cofunction identity: p sin u = cos a – u b . 2 Regroup.

= cos B a = cos a

p p – a b cos b + sin a – a b sin b Use the difference formula 2 2
for cosines. Use cofunction identities.

= sin a cos b + cos a sin b .

Thus, the sine of the sum of two angles equals the sine of the first angle times the cosine of the second angle plus the cosine of the first angle times the sine of the second angle. The final identity in the box, sin 1a – b 2 = sin a cos b – cos a sin b , gives a formula for sin1a – b 2, the sine of the difference of two angles. It is proved by writing sin1a – b 2 as sin3a + 1 – b 24 and then using the formula for the sine of a sum.

EXAMPLE 4

Using the Sine of a Sum to Find an Exact Value
7p p p 7p = + . using the fact that 12 12 3 4

Find the exact value of sin

Solution We apply the formula for the sine of a sum.
sin 7p p p = sin a + b 12 3 4 = sin p p p p cos + cos sin 3 4 3 4
sin1a + b 2 = sin a cos b + cos a sin b Substitute exact values.

23 # 22 1 22 + # 2 2 2 2 26 + 22 4

Simplify.

Check Point

Find the exact value of sin

5p using the fact that 12 p p 5p = + . 12 6 4

Section 6.2 Sum and Difference Formulas

621

EXAMPLE 5

Finding Exact Values
3 5

Suppose that sin a = 12 13 for a quadrant II angle a and sin b = angle b . Find the exact value of each of the following: a. cos a b. cos b

for a quadrant I

Solution
y

c. cos1a + b 2

d. sin1a + b 2.

a. We find cos a using a sketch that illustrates sin a = y 12 = . r 13

r = 13 y = 12 a x
12 13 :

Figure 6.2 shows a quadrant II angle a with sin a = 12 13 . We find x using x 2 + y2 = r2. Because a lies in quadrant II, x is negative. x2 + 12 2 = 132 x2 + 144 = 169
a lies in

x2 + y2 = r2 Square 12 and 13, respectively. Subtract 144 from both sides. Choose x = – 225 because in quadrant II, x is negative.

Figure 6.2 sin a =
quadrant II.

x = 25

x = – 225 = – 5 If x2 = 25, then x = ; 225 = ; 5.

Thus, cos a =
y

x -5 5 = = – . r 13 13 y 3 = . r 5

b. We find cos b using a sketch that illustrates sin b =
r=5 b x x

y=3

Figure 6.3 shows a quadrant I angle b with sin b = 3 5 . We find x using x2 + y2 = r2. x 2 + 3 2 = 52 x + 9 = 25 x = 16 x = 216 = 4
2 2

x2 + y2 = r2 Square 3 and 5, respectively. Subtract 9 from both sides. If x2 = 16, then x = ; 216 = ; 4. Choose x = 216 because in quadrant 1, x is positive.

Figure 6.3 sin b = 3 5 : b lies in
quadrant I.

Thus, cos b = 4 x = . r 5

We use the given values and the exact values that we determined to find exact values for cos1a + b 2 and sin1a + b 2.
These values are given. These are the values we found.

12 3 sin a= , sin b= 13 5

cos a=–

5 4 , cos b= 13 5

c. We use the formula for the cosine of a sum. cos1a + b 2 = cos a cos b – sin a sin b = a5 4 12 3 56 ba b a b = 13 5 13 5 65

622 Chapter 6 Analytic Trigonometry
d. We use the formula for the sine of a sum.
These values are given. These are the values we found.

sin1a + b 2 = sin a cos b + cos a sin b = 12 # 4 5 3 33 + a- b # = 13 5 13 5 65
1 2

12 3 sin a= , sin b= 13 5

cos a=–

5 4 , cos b= 13 5

Check Point
a. cos a

Suppose that sin a = 4 5 for a quadrant II angle a and sin b = for a quadrant I angle b . Find the exact value of each of the following: b. cos b c. cos1a + b 2 d. sin1a + b 2.

EXAMPLE 6

Verifying Observations on a Graphing Utility

3p p b in a c 0, 2p, d by 3 – 2, 2, 14 Figure 6.4 shows the graph of y = sin a x 2 2 viewing rectangle. a. Describe the graph using another equation. b. Verify that the two equations are equivalent.

Solution
Figure 6.4 The graph of
3p p b in a c 0, 2p, d 2 2 by 3 – 2, 2, 14 viewing rectangle y = sin a x –

a. The graph appears to be the cosine curve y = cos x. It cycles through maximum, intercept, minimum, intercept, and back to maximum. Thus, y = cos x also describes the graph. b. We must show that sin a x 3p b = cos x. 2

We apply the formula for the sine of a difference on the left side. sin a x 3p 3p 3p b = sin x cos – cos x sin 2 2 2 = sin x # 0 – cos x1 – 12 = cos x
sin1a – b 2 = sin a cos b – cos a sin b cos 3p 3p = 0 and sin = -1 2 2

Simplify.

This verifies our observation that y = sin a x the same graph.

3p b and y = cos x describe 2

Check Point

by 3 – 2, 2, 14 viewing rectangle.
Figure 6.5

Figure 6.5 shows the graph of y = cos a x +

3p p b in a c 0, 2p, d 2 2

a. Describe the graph using another equation. b. Verify that the two equations are equivalent.

Use sum and difference formulas for tangents.

Sum and Difference Formulas for Tangents

By writing tan1a + b 2 as the quotient of sin1a + b 2 and cos1a + b 2, we can develop a formula for the tangent of a sum. Writing subtraction as addition of an inverse leads to a formula for the tangent of a difference.

Section 6.2 Sum and Difference Formulas

623

Discovery
Derive the sum and difference formulas for tangents by working Exercises 55 and 56 in Exercise Set 6.2.

Sum and Difference Formulas for Tangents
tan1a + b 2 = tan a + tan b 1 – tan a tan b

The tangent of the sum of two angles equals the tangent of the first angle plus the tangent of the second angle divided by 1 minus their product. tan1a – b 2 = tan a – tan b 1 + tan a tan b

The tangent of the difference of two angles equals the tangent of the first angle minus the tangent of the second angle divided by 1 plus their product.

EXAMPLE 7
Verify the identity:

Verifying an Identity
tan a x p tan x – 1 b = . 4 tan x + 1

Solution We work with the left side.
p tan a x – b = 4 p 4 p 1 + tan x tan 4 tan x – 1 = 1 + tan x # 1 tan x – 1 = tan x + 1 tan x – tan
tan1a – b 2 = p = 1 4 tan a – tan b 1 + tan a tan b

tan

Check Point

Verify the identity:

tan1x + p2 = tan x.

Exercise Set 6.2
Practice Exercises
Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. 1. cos145° – 30°2 3. cos a 3p p – b 4 6 2. cos1120° – 45°2 4. cos a p 2p – b 3 6 In Exercises 9–12, verify each identity. 9. cos1a – b 2 cos a sin b cos1a – b 2 sin a sin b = tan a + cot b 7. cos 8. cos 5p p 5p p cos + sin sin 12 12 12 12 5p p 5p p cos + sin sin 18 9 18 9

In Exercises 5–8, each expression is the right side of the formula for cos1a – b 2 with particular values for a and b . a. Identify a and b in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. 5. cos 50° cos 20° + sin 50° sin 20° 6. cos 50° cos 5° + sin 50° sin 5°

10.

= cot a cot b + 1

11. cos a x 12. cos a x –

p 22 b = 1cos x + sin x2 4 2 22 5p b = 1cos x + sin x2 4 2

624 Chapter 6 Analytic Trigonometry
Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. 13. sin145° – 30°2 15. sin 105° 17. cos1135° + 30°2 19. cos 75° 21. tan a 23. tan a p p + b 6 4 4p p – b 3 4 14. sin160° – 45°2 16. sin 75° 18. cos1240° + 45°2 20. cos 105° 22. tan a 24. tan a p p + b 3 4 5p p – b 3 4 50. 48. cos1a + b 2 cos1a – b 2 h sin1x + h2 – sin x h = 1 – tan a tan b 1 + tan a tan b = cos x cos h – 1 sin h – sin x h h

49.

cos1x + h2 – cos x

= cos x

sin h cos h – 1 + sin x h h

51. sin 2a = 2 sin a cos a

Hint: Write sin 2a as sin1a + a2. Hint: Write cos 2a as cos1a + a2. 2 tan a 1 – tan2 a

52. cos 2a = cos2 a – sin2 a

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. 25. sin 25° cos 5° + cos 25° sin 5° 26. sin 40° cos 20° + cos 40° sin 20° 27. tan 10° + tan 35° 1 – tan 10° tan 35° 5p p 5p p cos – cos sin 12 4 12 4 7p p 7p p cos – cos sin 12 12 12 12 p 4p + tan 5 5 32. p 4p 1 – tan tan 5 5 tan 28. tan 50° – tan 20° 1 + tan 50° tan 20°

53. tan 2a =

Hint: Write tan 2a as tan1a + a2. 54. tan a p p + a b – tan a – a b = 2 tan 2a 4 4

Hint: Use the result in Exercise 53. 55. Derive the identity for tan1a + b 2 using tan1a + b 2 = cos1a + b 2 sin1a + b 2 .

29. sin 30. sin

p p – tan 5 30 31. p p 1 + tan tan 5 30 tan

After applying the formulas for sums of sines and cosines, divide the numerator and denominator by cos a cos b . 56. Derive the identity for tan1a – b 2 using tan1a – b 2 = tan3a + 1 – b 24.

In Exercises 33–54, verify each identity. 33. sin a x + 35. cos a x p b = cos x 2 p b = sin x 2 34. sin a x + 3p b = – cos x 2

After applying the formula for the tangent of the sum of two angles, use the fact that the tangent is an odd function. In Exercises 57–64, find the exact value of the following under the given conditions: a. cos1a + b 2
3 5,

36. cos1p – x2 = – cos x 38. tan1p – x2 = – tan x

37. tan12p – x2 = – tan x

39. sin1a + b 2 + sin1a – b 2 = 2 sin a cos b sin1a – b 2 cos a cos b sin1a + b 2 cos a cos b

b. sin1a + b 2

c. tan1a + b 2.
5 13 ,

40. cos1a + b 2 + cos1a – b 2 = 2 cos a cos b 41. 42. = tan a – tan b = tan a + tan b

57. sin a = a lies in quadrant I, and sin b = quadrant II. 58. sin a = 4 5 , a lies in quadrant I, and sin b = quadrant II.

b lies in b lies in

7 25 ,

1 59. tan a = – 3 4 , a lies in quadrant II, and cos b = 3 , b lies in quadrant I. 2 60. tan a = – 4 3 , a lies in quadrant II, and cos b = 3 , b lies in quadrant I. 8 61. cos a = 17 , a lies in quadrant IV, and sin b = – 1 2 , b lies in quadrant III. 1 62. cos a = 1 2 , a lies in quadrant IV, and sin b = – 3 , b lies in quadrant III.

p cos u + sin u 43. tan a u + b = 4 cos u – sin u 44. tan a cos u – sin u p – ub = 4 cos u + sin u

45. cos1a + b 2 cos1a – b 2 = cos2 b – sin2 a 46. sin1a + b 2 sin1a – b 2 = cos2 b – cos2 a 47. sin1a – b 2 sin1a + b 2 = tan a + tan b tan a – tan b

63. tan a = 3 4, p 6 a 6

3p 2 ,

3p and cos b = 1 4 , 2 6 b 6 2p. 3p 2 .

p 3 64. sin a = 5 6 , 2 6 a 6 p, and tan b = 7 , p 6 b 6

Section 6.2 Sum and Difference Formulas
In Exercises 65–68, the graph with the given equation is shown in a p c 0, 2p, d by 3 – 2, 2, 14 viewing rectangle. 2 a. Describe the graph using another equation. b. Verify that the two equations are equivalent. 65. y = sin1p – x2 – sin1a – b 2 + sin1a + b 2 cos1a – b 2 + cos1a + b 2

625

72.

73. cos a

p p p p + a b cos a – a b – sin a + a b sin a – a b 6 6 6 6

(Do not use four different identities to solve this exercise.) 74. sin a p p p p – a b cos a + a b + cos a – a b sin a + a b 3 3 3 3

(Do not use four different identities to solve this exercise.) In Exercises 75–78, half of an identity and the graph of this half are given. Use the graph to make a conjecture as to what the right side of the identity should be. Then prove your conjecture. 75. cos 2x cos 5x + sin 2x sin 5x = ? 66. y = cos1x – 2p2

[-2p, 2p, q] by [-2, 2, 1]

76. sin 5x cos 2x – cos 5x sin 2x = ? p p 67. y = sin a x + b + sin a – x b 2 2

[-2p, 2p, q] by [-2, 2, 1]

77. sin 68. y = cos a x p p b – cos a x + b 2 2

5x 5x cos 2x – cos sin 2x = ? 2 2

[-2p, 2p, q] by [-2, 2, 1]

78. cos

Practice Plus
In Exercises 69–74, rewrite each expression as a simplified expression containing one term. 69. cos1a + b 2 cos b + sin1a + b 2 sin b 70. sin1a – b 2 cos b + cos1a – b 2 sin b 71. cos1a + b 2 + cos1a – b 2 sin1a + b 2 – sin1a – b 2

5x 5x cos 2x + sin sin 2x = ? 2 2

[-2p, 2p, q] by [-2, 2, 1]

626 Chapter 6 Analytic Trigonometry

Application Exercises
79. A ball attached to a spring is raised 2 feet and released with an initial vertical velocity of 3 feet per second. The distance of the ball from its rest position after t seconds is given by d = 2 cos t + 3 sin t. Show that 2 cos t + 3 sin t = 213 cos1t – u2, where u lies in quadrant I and tan u = 3 2 . Use the identity to find the amplitude and the period of the ball’s motion. 80. A tuning fork is held a certain distance from your ears and struck. Your eardrums’ vibrations after t seconds are given by p = 3 sin 2t. When a second tuning fork is struck, the formula p = 2 sin12t + p2 describes the effects of the sound on the eardrums’ vibrations. The total vibrations are given by p = 3 sin 2t + 2 sin12t + p2. a. Simplify p to a single term containing the sine. b. If the amplitude of p is zero, no sound is heard. Based on your equation in part (a), does this occur with the two tuning forks in this exercise? Explain your answer.

90. sin a x + 91. cos a x +

p p b = sin x + sin 2 2 p p b = cos x + cos 2 2

92. cos 1.2x cos 0.8x – sin 1.2x sin 0.8x = cos 2x 93. sin 1.2x cos 0.8x + cos 1.2x sin 0.8x = sin 2x

Critical Thinking Exercises
Make Sense? In Exercises 94–97, determine whether each statement makes sense or does not make sense, and explain your reasoning.
94. I’ve noticed that for sine, cosine, and tangent, the trig function for the sum of two angles is not equal to that trig function of the first angle plus that trig function of the second angle. 95. After using an identity to determine the exact value of sin 105°, I verified the result with a calculator. 96. Using sum and difference formulas, I can find exact values for sine, cosine, and tangent at any angle.

Writing in Mathematics
In Exercises 81–86, use words to describe the formula for each of the following: 81. the cosine of the difference of two angles. 82. the cosine of the sum of two angles. 83. the sine of the sum of two angles. 84. the sine of the difference of two angles. 85. the tangent of the difference of two angles. 86. the tangent of the sum of two angles. 87. The distance formula and the definitions for cosine and sine are used to prove the formula for the cosine of the difference of two angles. This formula logically leads the way to the other sum and difference identities. Using this development of ideas and formulas, describe a characteristic of mathematical logic.

97. After the difference formula for cosines is verified, I noticed that the other sum and difference formulas are verified relatively quickly. 98. Verify the identity: sin1x – y2 cos x cos y sin1y – z2 cos y cos z sin1z – x2 cos z cos x = 0.

In Exercises 99–102, find the exact value of each expression. Do not use a calculator. 99. sin a cos-1 100. sin B sin-1 1 3 + sin-1 b 2 5 3 4 – cos-1 a – b R 5 5 4 5 + cos-1 b 3 13 23 1 = – sin-1 a – b R 2 2

101. cos a tan-1

Technology Exercises
In Exercises 88–93, graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of x for which both sides are defined but not equal. 3p 88. cos a – x b = – sin x 2 89. tan1p – x2 = – tan x

102. cos B cos-1 ¢ –

In Exercises 103–105, write each trigonometric expression as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function. 103. cos1sin-1 x – cos-1 y2 104. sin1tan-1 x – sin-1 y2 105. tan1sin-1 x + cos-1 y2

Section 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas

627

Group Exercise
106. Remembering the six sum and difference identities can be difficult. Did you have problems with some exercises because the identity you were using in your head turned out to be an incorrect formula? Are there easy ways to remember the six new identities presented in this section? Group members should address this question, considering one identity at a time. For each formula, list ways to make it easier to remember.

107. Give exact values for sin 30°, cos 30°, sin 60°, and cos 60°. 108. Use the appropriate values from Exercise 107 to answer each of the following. a. Is sin 2 # 30°, or sin 60°, equal to 2 sin 30°? b. Is sin 2 # 30°, or sin 60°, equal to 2 sin 30° cos 30°? 109. Use appropriate values from Exercise 107 to answer each of the following. a. Is cos (2 # 30°), or cos 60°, equal to 2 cos 30°? b. Is cos (2 # 30°), or cos 60°, equal to cos2 30° – sin2 30°?

Preview Exercises
Exercises 107–109 will help you prepare for the material covered in the next section.

Section Objectives ? Use the double-angle ?
formulas. Use the power-reducing formulas.

6.3

Double-Angle, Power-Reducing, and Half-Angle Formulas
e have a long history of throwing things. Prior to 400 B.C., the Greeks competed in games that included discus throwing. In the seventeenth century, English soldiers organized cannonball-throwing competitions. In 1827, a Yale University student, disappointed over failing an exam, took out his frustrations at the passing of a collection plate in chapel. Seizing the monetary tray, he flung it in the direction of a large open space on campus. Yale students see this act of frustration as the origin of the Frisbee. In this section, we develop other important classes of identities, called the doubleangle, power-reducing, and half-angle formulas. We will see how one of these formulas can be used by athletes to increase throwing distance.

? Use the half-angle formulas.

Use the double-angle formulas.

Double-Angle Formulas
A number of basic identities follow from the sum formulas for sine, cosine, and tangent. The first category of identities involves double-angle formulas.

Double-Angle Formulas
sin 2u = 2 sin u cos u cos 2u = cos2 u – sin2 u tan 2u = 2 tan u 1 – tan2 u

To prove each of these formulas, we replace a and b by u in the sum formulas for sin1a + b 2, cos1a + b 2, and tan1a + b 2.

Section 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas

627

Preview Exercises
Exercises 107–109 will help you prepare for the material covered in the next section.

Section Objectives ? Use the double-angle ?
formulas. Use the power-reducing formulas.

6.3

? Use the half-angle formulas.

Use the double-angle formulas.

Double-Angle Formulas
A number of basic identities follow from the sum formulas for sine, cosine, and tangent. The first category of identities involves double-angle formulas.

Double-Angle Formulas
sin 2u = 2 sin u cos u cos 2u = cos2 u – sin2 u tan 2u = 2 tan u 1 – tan2 u

To prove each of these formulas, we replace a and b by u in the sum formulas for sin1a + b 2, cos1a + b 2, and tan1a + b 2.

628 Chapter 6 Analytic Trigonometry Study Tip
The 2 that appears in each of the double-angle expressions cannot be pulled to the front and written as a coefficient. Incorrect! sin 2u = 2 sin u cos 2u = 2 cos u tan 2u = 2 tan u The figure shows that the graphs of y = sin 2x and y = 2 sin x do not coincide: sin 2x Z 2 sin x.
y = 2 sin x y = sin 2x

• sin 2u=sin (u+u)=sin u cos u+cos u sin u=2 sin u cos u
We use sin (a + b) = sin a cos b + cos a sin b.

• cos 2u=cos (u+u)=cos u cos u-sin u sin u=cos2 u-sin2 u
We use cos (a + b) = cos a cos b – sin a sin b.

• tan 2u=tan (u+u)=

tan u+tan u 2 tan u = 1-tan u tan u 1-tan2 u

We use tan a + tan b tan (a + b) = . 1 – tan a tan b

EXAMPLE 1
If sin u =
5 13

Using Double-Angle Formulas to Find Exact Values

and u lies in quadrant II, find the exact value of each of the following: b. cos 2u c. tan 2u.

a. sin 2u

Solution We begin with a sketch that illustrates
[0, 2p, q] by [-3, 3, 1]

sin u =

y 5 = . r 13
5 13 .

y (-12, 5) y=5 r = 13 u O
5 13

Figure 6.6 shows a quadrant II angle u for which sin u = x2 + y2 = r2. Because u lies in quadrant II, x is negative. x 2 + 52 x2 + 25 x2 x = = = = 132 169 144 – 2144 = – 12
x2 + y2 = r2

We find x using

Square 5 and 13, respectively. Subtract 25 from both sides. If x2 = 144, then x = ; 2144 = ; 12. Choose x = – 2144 because in quadrant II, x is negative.

Figure 6.6 sin u =
quadrant II.

and u lies in

Now we can use values for x, y, and r to find the required values. We will use y x 12 5 5 cos u = = = – . We were given sin u = . and tan u = r x 13 12 13 a. sin 2u = 2 sin u cos u = 2 a 5 12 120 b a- b = 13 13 169 12 2 5 2 144 25 119 b – a b = = 13 13 169 169 169 5 b 12 –

b. cos 2u = cos2 u – sin2 u = a2 tan u = c. tan 2u = 1 – tan2 u 2a –

5 5 6 6 = = 2 25 119 5 1 1 – a- b 144 144 12 5 144 120 b = = a- b a 6 119 119
4 5

Check Point
a. sin 2u

If sin u =

and u lies in quadrant II, find the exact value of each

of the following: b. cos 2u c. tan 2u.

Section 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas

629

EXAMPLE 2

Using the Double-Angle Formula for Tangent to Find an Exact Value
2 tan 15° . 1 – tan2 15°

Find the exact value of

Solution The given expression is the right side of the formula for tan 2u with
u = 15°.
u tan 2u = 2 tan 2 1 – tan u

2 tan 15? ?3 =tan(2 ? 15? )=tan 30? = 1-tan2 15? 3

Check Point

Find the exact value of cos2 15° – sin2 15°.

There are three forms of the double-angle formula for cos 2u. The form we have seen involves both the cosine and the sine: cos 2u = cos2 u – sin2 u. There are situations where it is more efficient to express cos 2u in terms of just one trigonometric function. Using the Pythagorean identity sin2 u + cos2 u = 1, we can write cos 2u = cos2 u – sin2 u in terms of the cosine only. We substitute 1 – cos2 u for sin2 u. cos 2u = cos2 u – sin2 u = cos2 u – 11 – cos2 u2 = cos2 u – 1 + cos2 u = 2 cos2 u – 1

We can also use a Pythagorean identity to write cos 2u in terms of sine only. We substitute 1 – sin2 u for cos2 u. cos 2u = cos2 u – sin2 u = 1 – sin2 u – sin2 u = 1 – 2 sin2 u

Three Forms of the Double-Angle Formula for cos 2U
cos 2u = cos2 u – sin2 u cos 2u = 2 cos2 u – 1 cos 2u = 1 – 2 sin2 u

EXAMPLE 3
Verify the identity:

Verifying an Identity
cos 3u = 4 cos3 u – 3 cos u.

Solution We begin by working with the left side. In order to obtain an expression for cos 3u, we use the sum formula and write 3u as 2u + u.
cos 3u = cos12u + u2
2 cos2 u – 1

Write 3u as 2u + u. cos1a + b 2 = cos a cos b – sin a sin b

=cos 2u cos u-sin 2u sin u
2 sin u cos u

= 12 cos2 u – 12 cos u – 2 sin u cos u sin u

Substitute double-angle formulas. Because the right side of the given equation involves cosines only, use this form for cos 2u. Multiply.

=2 cos3 u-cos u-2 sin2 u cos u
1 – cos2 u

630 Chapter 6 Analytic Trigonometry
cos 3u = 2 cos3 u – cos u – 2 sin2 u cos u = 2 cos3 u – cos u – 211 – cos2 u2 cos u = 2 cos3 u – cos u – 2 cos u + 2 cos3 u = 4 cos u – 3 cos u
3

We’ve repeated the last step from the previous page. To get cosines only, use sin2 u + cos2 u = 1 and substitute 1 – cos2 u for sin2 u. Multiply. Simplify: 2 cos3 u + 2 cos3 u = 4 cos3 u and – cos u – 2 cos u = – 3 cos u.

We were required to verify cos 3u = 4 cos3 u – 3 cos u. By working with the left side, cos 3u, and expressing it in a form identical to the right side, we have verified the identity.

Check Point

Verify the identity:

sin 3u = 3 sin u – 4 sin3 u.

Use the power-reducing formulas.

Power-Reducing Formulas
The double-angle formulas are used to derive the power-reducing formulas:

Power-Reducing Formulas
sin2 u = 1 – cos 2u 2 cos2 u = 1 + cos 2u 2 tan2 u = 1 – cos 2u 1 + cos 2u

We can prove the first two formulas in the box by working with two forms of the double-angle formula for cos 2u.
This is the form with sine only. This is the form with cosine only.

cos 2u=1-2 sin2 u

cos 2u=2 cos2 u-1

Solve the formula on the left for sin2 u. Solve the formula on the right for cos2 u. 2 sin2 u = 1 – cos 2u sin2 u = 1 – cos 2u 2 2 cos2 u = 1 + cos 2u cos2 u = 1 + cos 2u 2
Divide both sides of each equation by 2.

These are the first two formulas in the box. The third formula in the box is proved by writing the tangent as the quotient of the sine and the cosine. 1 – cos 2u 1 2 sin u 2 1 – cos 2u # 2 1 – cos 2u 2 tan u = = = = 2 1 + cos 2u 2 1 + cos 2u 1 + cos 2u cos u 1 2 Power-reducing formulas are quite useful in calculus. By reducing the power of trigonometric functions, calculus can better explore the relationship between a function and how it is changing at every single instant in time.

EXAMPLE 4

Reducing the Power of a Trigonometric Function

Write an equivalent expression for cos4 x that does not contain powers of trigonometric functions greater than 1.

Section 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas

631

Solution We will apply the formula for cos2 u twice.
cos4 x = 1cos2 x2 = a
2

1 + cos 2x 2 b 2 1 + 2 cos 2x + cos2 2x = 4

Use cos2 u =

1 + cos 2u with u = x. 2

Square the numerator: 1A + B22 = A2 + 2AB + B2. Square the denominator.

1 1 1 = + cos 2x+ cos2 2x 4 2 4
We can reduce the power of cos2 2x using 2 u = 1 + cos 2u cos 2 with u = 2x.

Divide each term in the numerator by 4.

1 1 1 1 + cos 212×2 + cos 2x + B R 4 2 4 2

Use the power-reducing formula for cos2 u with u = 2x. Multiply.
1 Distribute 8 throughout parentheses.

1 1 1 + cos 2x + 11 + cos 4×2 4 2 8 1 1 1 1 = + cos 2x + + cos 4x 4 2 8 8 3 1 1 = + cos 2x + cos 4x 8 2 8

1 Simplify: 4 +

1 8

2 8

1 8

3 8.

Use the half-angle formulas.

1 1 4 Thus, cos4 x = 3 8 + 2 cos 2x + 8 cos 4x. The expression for cos x does not contain powers of trigonometric functions greater than 1.

Study Tip
The 1 2 that appears in each of the halfangle formulas cannot be pulled to the front and written as a coefficient. Incorrect! sin cos 1 u = sin u 2 2

Check Point

Write an equivalent expression for sin4 x that does not contain powers of trigonometric functions greater than 1.

Half-Angle Formulas
Useful equivalent forms of the power-reducing formulas can be obtained by replacing a u with . Then solve for the trigonometric function on the left sides of the equations. 2 The resulting identities are called the half-angle formulas:

u 1 = cos u 2 2 u 1 tan = tan u 2 2 The figure shows that the graphs of x 1 and y = sin x do not y = sin 2 2 coincide: sin x 1 Z sin x. 2 2
1 y=2 sin x

Half-Angle Formulas
sin a 1 – cos a = ; 2 A 2 a 1 + cos a = ; 2 A 2 a 1 – cos a = ; 2 A 1 + cos a

cos
y = sin x 2

tan

[0, 2p, q] by [-2, 2, 1]

The ; symbol in each formula does not mean that there are two possible values for each function. Instead, the ; indicates that you must determine the sign of the trigonometric function, + or – , based on the quadrant in which the a half-angle lies. 2

632 Chapter 6 Analytic Trigonometry
If we know the exact value for the cosine of an angle, we can use the half-angle formulas to find exact values of sine, cosine, and tangent for half of that angle. For 22 . In the next example, we find the exact 2 value of the cosine of half of 225°, or cos 112.5°. example, we know that cos 225° = –

EXAMPLE 5

Using a Half-Angle Formula to Find an Exact Value

Find the exact value of cos 112.5°. 225° a , we use the half-angle formula for cos with 2 2 a = 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° 6 0. Thus, we use the – sign in the half-angle formula.

Solution Because 112.5° =

cos 112.5° = cos

225° 2 1 + cos 225° 2 1 + ¢22 = 2
cos 225° = Use cos a 1 + cos a = with a = 225°. 2 A 2

= –

Discovery
Use your calculator approximations for to find

22 2

= –

2 – 22 C 4

Multiply the radicand by 2 2: 1 + ¢22 = 2 # 2 = 2 – 22 . 2 2 4

32 – 22 2 and cos 112.5°. What do you observe?

= –

32 – 22 2

Simplify: 24 = 2.

Study Tip
Keep in mind as you work with the half-angle formulas that the sign outside the radical is a determined by the half angle . By contrast, the sign of cos a, which appears under the 2 radical, is determined by the full angle a.

sin

1-cos a a =— 2 A 2

The sign is determined by the quadrant of a . 2

The sign of cos a is determined by the quadrant of a.

Check Point

Use cos 210° = –

23 to find the exact value of cos 105°. 2

a that do not require us to determine 2 what sign to use when applying the formula. These formulas are logically connected to the identities in Example 6 and Check Point 6. There are alternate formulas for tan

Section 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas

633

EXAMPLE 6
Verify the identity:

Verifying an Identity
tan u = 1 – cos 2u . sin 2u
The form cos 2u = 1 – 2 sin2 u is used because it produces only one term in the numerator. Use the double-angle formula for sine in the denominator. Simplify the numerator. Divide the numerator and denominator by 2 sin u. Use a quotient identity: tan u = sin u . cos u

Solution We work with the right side.
1 – 11 – 2 sin2 u2 1 – cos 2u = sin 2u 2 sin u cos u = 2 sin2 u 2 sin u cos u sin u = cos u = tan u

The right side simplifies to tan u, the expression on the left side. Thus, the identity is verified.

Check Point

6 Verify the identity:

tan u =

sin 2u . 1 + cos 2u

a Half-angle formulas for tan can be obtained using the identities in Example 6 2 and Check Point 6: tan u = 1 – cos 2u sin 2u and tan u = sin 2u . 1 + cos 2u

a Do you see how to do this? Replace each occurrence of u with . This results in the 2 following identities:

Half-Angle Formulas for Tangent
tan tan a 1 – cos a = 2 sin a a sin a = 2 1 + cos a

EXAMPLE 7
Verify the identity:

Verifying an Identity
tan a = csc a – cot a. 2

Solution We begin with the right side.
csc a-cot a= 1 cos a 1-cos a a = =tan sin a sin a sin a 2
This is the first of the two half-angle formulas in the preceding box.

Express functions in terms of sines and cosines.

We worked with the right side and arrived at the left side. Thus, the identity is verified.

Check Point

Verify the identity:

tan

sec a a = . 2 sec a csc a + csc a

634 Chapter 6 Analytic Trigonometry
We conclude with a summary of the principal trigonometric identities developed in this section and the previous section. The fundamental identities can be found in the box on page 606.

Principal Trigonometric Identities
Sum and Difference Formulas sin1a + b 2 = sin a cos b + cos a sin b sin1a – b 2 = sin a cos b – cos a sin b

cos1a + b 2 = cos a cos b – sin a sin b tan1a + b 2 = tan a + tan b 1 – tan a tan b

cos1a – b 2 = cos a cos b + sin a sin b tan1a – b 2 = tan a – tan b 1 + tan a tan b

Double-Angle Formulas sin 2u = 2 sin u cos u cos 2u = cos2 u – sin2 u = 2 cos2 u – 1 = 1 – 2 sin2 u tan 2u = 2 tan u 1 – tan2 u

Power-Reducing Formulas

Study Tip
To help remember the correct sign in the numerator in the first two power-reducing formulas and the first two half-angle formulas, remember sinus-minus—the sine is minus.

sin2 u =

1 – cos 2u 2

cos2 u =

1 + cos 2u 2

tan2 u =

1 – cos 2u 1 + cos 2u

Half-Angle Formulas sin a 1 – cos a = ; 2 A 2 cos a 1 + cos a = ; 2 A 2

tan

a 1 – cos a 1 – cos a sin a = ; = = 2 A 1 + cos a sin a 1 + cos a

Exercise Set 6.3
Practice Exercises
In Exercises 1–6, use the figures to find the exact value of each trigonometric function. In Exercises 7–14, use the given information to find the exact value of each of the following: a. sin 2u 7. sin u = 8. sin u =
5 3 u 4 a 24 7

b. cos 2u

c. tan 2u.

9. cos u = 10. cos u =

15 17 , u lies in quadrant II. 12 13 , u lies in quadrant II. 24 25 , u lies in quadrant IV. 40 41 , u lies in quadrant IV.

11. cot u = 2, u lies in quadrant III. 12. cot u = 3, u lies in quadrant III.
9 13. sin u = – 41 , u lies in quadrant III.

1. sin 2u 4. sin 2a

2. cos 2u 5. cos 2a

3. tan 2u 6. tan 2a

14. sin u = – 2 3 , u lies in quadrant III.

Section 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas
In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 15. 2 sin 15° cos 15° 17. cos 75° – sin 75° p 19. 2 cos2 – 1 8 2 tan 21. p 12 p 12
2 2

635

16. 2 sin 22.5° cos 22.5° 18. cos2 105° – sin2 105° p 20. 1 – 2 sin2 12 2 tan 22. p 8 p 8

In Exercises 55–58, use the given information to find the exact value of each of the following: a a a a. sin b. cos c. tan . 2 2 2 55. tan a = 4 3 , 180° 6 a 6 270° 56. tan a =
8 15 ,

180° 6 a 6 270°

p 57. sec a = – 13 5 , 2 6 a 6 p

1 – tan2

58. sec a = – 3, p 2 6 a 6 p In Exercises 59–68, verify each identity.

In Exercises 23–34, verify each identity. 2 tan u 23. sin 2u = 1 + tan2 u 25. 1sin u + cos u22 = 1 + sin 2u 26. 1sin u – cos u22 = 1 – sin 2u 27. sin2 x + cos 2x = cos2 x 28. 1 – tan2 x = 30. cot x = cos 2x cos2 x 29. cot x = sin 2x 1 – cos 2x 63. tan 2 cot u 24. sin 2u = 1 + cot2 u 59. sin2

u sec u – 1 = 2 2 sec u u sin u + tan u = 2 2 tan u

60. sin2

u csc u – cot u = 2 2 csc u u sec u + 1 = 2 2 sec u

61. cos2

62. cos2

a tan a = 2 sec a + 1 a sin2 a + 1 – cos2 a = 2 sin a11 + cos a2 66. cot x 1 + cos x = 2 sin x x x – cot = – 2 cot x 2 2

1 + cos 2x sin 2x

64. 2 tan

31. sin 2t – tan t = tan t cos 2t 32. sin 2t – cot t = – cot t cos 2t 33. sin 4t = 4 sin t cos3 t – 4 sin3 t cos t 34. cos 4t = 8 cos4 t – 8 cos2 t + 1 In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 35. 6 sin4 x 37. sin x cos x
2 2

65. cot

x sin x = 2 1 – cos x x x + cot = 2 csc x 2 2

67. tan

68. tan

Practice Plus
In Exercises 69–78, half of an identity and the graph of this half are given. Use the graph to make a conjecture as to what the right side of the identity should be. Then prove your conjecture. 69. cot x – tan x = ? cot x + tan x

36. 10 cos4 x 38. 8 sin2 x cos2 x

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. 39. sin 15° 42. sin 105° 45. tan 7p 8 40. cos 22.5° 43. tan 75° 46. tan 3p 8 41. cos 157.5° 44. tan 112.5°

In Exercises 47–54, use the figures to find the exact value of each trigonometric function.
5 3 u 4 a 24 7

[-2p, 2p, q] by [-3, 3, 1]

70.

21tan x – cot x2 tan2 x – cot2 x

= ?

u 2 a 50. sin 2 47. sin u u 53. 2 sin cos 2 2

u 2 a 51. cos 2 a a 54. 2 sin cos 2 2 48. cos

u 2 a 52. tan 2 49. tan
[-2p, 2p, q] by [-3, 3, 1]

636 Chapter 6 Analytic Trigonometry
71. a sin x x 2 + cos b = ? 2 2 76. tan x + cot x = ?

[-2p, 2p, q] by [-3, 3, 1] [-2p, 2p, q] by [-3, 3, 1]

77. sin x14 cos2 x – 12 = ?

72. sin2

x x – cos2 = ? 2 2

[0, 2p, k] by [-3, 3, 1] [-2p, 2p, q] by [-3, 3, 1]

78. 1 – 8 sin2 x cos2 x = ?

73.

sin 2x cos 2x = ? cos x sin x

[0, 2p, k ] by [-3, 3, 1] 8

[-2p, 2p, q] by [-3, 3, 1]

Application Exercises
79. Throwing events in track and field include the shot put, the discus throw, the hammer throw, and the javelin throw. The distance that the athlete can achieve depends on the initial speed of the object thrown and the angle above the horizontal at which the object leaves the hand. This angle is represented by u in the figure shown. The distance, d, in feet, that the athlete throws is modeled by the formula v2 0 d = sin u cos u, 16 in which v0 is the initial speed of the object thrown, in feet per second, and u is the angle, in degrees, at which the object leaves the hand.

74. sin 2x sec x = ?

[-2p, 2p, q] by [-3, 3, 1]

75.

csc2 x = ? cot x

a. Use an identity to express the formula so that it contains the sine function only. b. Use your formula from part (a) to find the angle, u, that produces the maximum distance, d, for a given initial speed, v0 .

[-2p, 2p, q] by [-3, 3, 1]

Section 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas
Use this information to solve Exercises 80–81: The speed of a supersonic aircraft is usually represented by a Mach number, named after Austrian physicist Ernst Mach (1838–1916). A Mach number is the speed of the aircraft, in miles per hour, divided by the speed of sound, approximately 740 miles per hour. Thus, a plane flying at twice the speed of sound has a speed, M, of Mach 2. 90. Explain how the double-angle formulas are derived.

637

91. How can there be three forms of the double-angle formula for cos 2u? 92. Without showing algebraic details, describe in words how to reduce the power of cos4 x. 93. Describe one or more of the techniques you use to help remember the identities in the box on page 634. 94. Your friend is about to compete as a shot-putter in a college field event. Using Exercise 79(b), write a short description to your friend on how to achieve the best distance possible in the throwing event.

Concord Mach 2.03

SR-71 Blackbird Mach 3.3

Technology Exercises
In Exercises 95–98, graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, find a value of x for which both sides are defined but not equal. x 95. 3 – 6 sin2 x = 3 cos 2x 96. 4 cos2 = 2 + 2 cos x 2 x 1 x 1 97. sin = sin x 98. cos = cos x 2 2 2 2 In Exercises 99–101, graph each equation in a c – 2p, 2p, p d by 2 3 – 3, 3, 14 viewing rectangle. Then a. Describe the graph using

If an aircraft has a speed greater than Mach 1, a sonic boom is heard, created by sound waves that form a cone with a vertex angle u, shown in the figure.
Sonic boom cone

The relationship between the cone’s vertex angle, u, and the Mach speed, M, of an aircraft that is flying faster than the speed of sound is given by 1 u . sin = 2 M p 80. If u = , determine the Mach speed, M, of the aircraft. 6 Express the speed as an exact value and as a decimal to the nearest tenth. 81. If u = p , determine the Mach speed, M, of the aircraft. 4 Express the speed as an exact value and as a decimal to the nearest tenth.

another equation, and b. Verify that the two equations are equivalent. x 2 tan 1 – 2 cos 2x 2 99. y = 100. y = 2 sin x – 1 x 1 + tan2 2 101. y = csc x – cot x

Critical Thinking Exercises
Make Sense? In Exercises 102–105, determine whether each statement makes sense or does not make sense, and explain your reasoning.
102. The double-angle identities are derived from the sum identities by adding an angle to itself. 103. I simplified a double-angle trigonometric expression by pulling 2 to the front and treating it as a coefficient. 104. When using the half-angle formulas for trigonometric a functions of , I determine the sign based on the quadrant 2 in which a lies. 105. I used a half-angle formula to find the exact value of cos 100°. 106. Verify the identity: sin3 x + cos3 x = 1sin x + cos x2a 1 sin 2x b. 2

Writing in Mathematics
In Exercises 82–89, use words to describe the formula for: 82. the sine of double an angle. 83. the cosine of double an angle. (Describe one of the three formulas.) 84. the tangent of double an angle. 85. the power-reducing formula for the sine squared of an angle. 86. the power-reducing formula for the cosine squared of an angle. 87. the sine of half an angle. 88. the cosine of half an angle. 89. the tangent of half an angle. (Describe one of the two formulas that does not involve a square root.)

In Exercises 107–110, find the exact value of each expression. Do not use a calculator. 107. sin ¢ 2 sin-1 23 = 2 4 108. cos B 2 tan-1 a – b R 3

638 Chapter 6 Analytic Trigonometry
1 3 109. cos2 a sin-1 b 2 5 1 3 110. sin2 a cos-1 b 2 5

Preview Exercises
Exercises 113–115 will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. 113. sin 60° sin 30° = 1 2 3cos 160° – 30°2 – cos 160° + 30°24 114. cos p p 1 p p p p cos = B cos a – b + cos a + b R 2 3 2 2 3 2 3 1 p p p = B sin a p + b + sin a p – b R 2 2 2 2

111. Use a right triangle to write sin 12 sin-1 x2 as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function. 112. Use the power-reducing formulas to rewrite sin x as an equivalent expression that does not contain powers of trigonometric functions greater than 1.
6

115. sin p cos

Chapter

Mid-Chapter Check Point

What you Know: Verifying an identity means showing that the expressions on each side are identical. Like solving
puzzles, the process can be intriguing because there are sometimes several “best” ways to proceed. We presented some guidelines to help you get started (see page 613). We used fundamental trigonometric identities (see page 606), as well as sum and difference formulas, double-angle formulas, power-reducing formulas, and half-angle formulas (see page 634) to verify identities. We also used these formulas to find exact values of trigonometric functions.

Study Tip
Make copies of the boxes on pages 606 and 634 that contain the essential trigonometric identities. Mount these boxes on cardstock and add this reference sheet to the one you prepared for Chapter 5. (If you didn’t prepare a reference sheet for Chapter 5, it’s not too late: See the study tip on page 601.) In Exercises 1–18, verify each identity. 1. cos x1tan x + cot x2 = csc x 2. sin1x + p2 3p cos ¢ x + = 2 3. 1sin u + cos u22 + 1sin u – cos u22 = 2 cos t – cot t sin t – 1 4. = cos t cos t cot t 5. 1 – cos 2x = tan x sin 2x sec2 x 2 – sec2 x tan2 a – tan2 b 1 – tan2 a tan2 b

14. sec 2x =

= tan2 x – sec2 x

15. tan1a + b 2 tan1a – b 2 = 16. csc u + cot u = 17. sin u 1 – cos u

1 2 tan x = csc 2x 1 + tan2 x

18.

sec t – 1 1 – cos t = t sec t t

Use the following conditions to solve Exercises 19–22: p 6 a 6 p 2 12 3p . cos b = – , p 6 b 6 13 2 sin a = Find the exact value of each of the following. 19. cos1a – b 2 21. sin 2a 20. tan1a + b 2 22. cos b 2 3 , 5

6. sin u cos u + cos2 u =

cos u11 + cot u2 csc u

sin x cos x 7. + = sin x + cos x tan x cot x 8. sin2 t tan t – sin t = 2 2 tan t 1 3sin1a + b 2 + sin1a – b 24 2

9. sin a cos b =

1 + csc x 10. – cot x = cos x sec x cot x – 1 1 – tan x 11. = cot x + 1 1 + tan x 12. 2 sin3 u cos u + 2 sin u cos3 u = sin 2u 13. sin t + cos t sin t = sec t + csc t sec t

In Exercises 23–26, find the exact value of each expression. Do not use a calculator. 23. sin a 25. cos 5p 3p + b 4 6 24. cos2 15° – sin2 15°

5p p 5p p cos + sin sin 12 12 12 12

26. tan 22.5°

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