Using Sum and Difference Formulas - Big Ideas Learning

[Pages:6]9.8

Using Sum and Difference Formulas

Essential Question How can you evaluate trigonometric

functions of the sum or difference of two angles?

Deriving a Difference Formula

Work with a partner. a. Explain why the two triangles shown are congruent.

(cos a, sin a) 1

yd

a b

(cos b, sin b)

x

y (cos(a - b), sin(a - b))

d 1

a - b (1, 0)

x

CONSTRUCTING VIABLE ARGUMENTS

To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.

b. Use the Distance Formula to write an expression for d in the first unit circle. c. Use the Distance Formula to write an expression for d in the second unit circle. d. Write an equation that relates the expressions in parts (b) and (c). Then simplify

this equation to obtain a formula for cos(a - b).

Deriving a Sum Formula Work with a partner. Use the difference formula you derived in Exploration 1 to write a formula for cos(a + b) in terms of sine and cosine of a and b. Hint: Use the fact that

cos(a + b) = cos[a - (-b)].

Deriving Difference and Sum Formulas

Work with a partner. Use the formulas you derived in Explorations 1 and 2 to write formulas for sin(a - b) and sin(a + b) in terms of sine and cosine of a and b. Hint: Use the cofunction identities

( ) ( ) sin --2 - a = cos a and cos --2 - a = sin a

and the fact that

[( ) ] cos --2 - a + b = sin(a - b) and sin(a + b) = sin[a - (-b)].

Communicate Your Answer

4. How can you evaluate trigonometric functions of the sum or difference of two angles?

5. a. Find the exact values of sin 75? and cos 75? using sum formulas. Explain your reasoning.

b. Find the exact values of sin 75? and cos 75? using difference formulas. Compare your answers to those in part (a).

Section 9.8 Using Sum and Difference Formulas 519

9.8 Lesson

Core Vocabulary

Previous ratio

What You Will Learn

Use sum and difference formulas to evaluate and simplify trigonometric expressions.

Use sum and difference formulas to solve trigonometric equations and rewrite real-life formulas.

Using Sum and Difference Formulas

In this lesson, you will study formulas that allow you to evaluate trigonometric functions of the sum or difference of two angles.

Core Concept

Sum and Difference Formulas Sum Formulas

sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b - sin a sin b

Difference Formulas sin(a - b) = sin a cos b - cos a sin b cos(a - b) = cos a cos b + sin a sin b

tan(a + b) = -- 1ta-n taan+-- atatannbb

tan(a - b) = -- 1ta+n taan--- atatannbb

In general, sin(a + b) sin a + sin b. Similar statements can be made for the other trigonometric functions of sums and differences.

Check sin(15?) .2588190451 ( (6)- (2))/4 .2588190451

Check tan(7/12) -3.732050808 -2- (3) -3.732050808

Evaluating Trigonometric Expressions Find the exact value of (a) sin 15? and (b) tan -- 712 .

SOLUTION

a. sin 15? = sin(60? - 45?)

Substitute 60? - 45? for 15?.

= sin 60? cos 45? - cos 60? sin 45?

Difference formula for sine

( ) ( ) -- --

--

= -- 23 -- 22 - --21 -- 22

Evaluate.

--

--

= -- 6 -4 2

Simplify.

--

--

The exact value of sin 15? is -- 6 -4 2 . Check this with a calculator.

( ) b. tan -- 712 = tan --3 + --4

= -- 1ta-n t--3an+-- --3tatann--4--4

Substitute --3 + --4 for -- 712 . Sum formula for tangent

--

=

3 + 1 -- 1 - --3 1

= -2 - --3

Evaluate. Simplify.

The exact value of tan -- 712 is -2 - --3. Check this with a calculator.

520 Chapter 9 Trigonometric Ratios and Functions

ANOTHER WAY

You can also use a Pythagorean identity and quadrant signs to find sin a and cos b.

Using a Difference Formula

Find cos(a - 0 < b < --2 .

b)

given

that

cos

a

=

---54

with

<

a

<

-- 32

and

sin

b

=

-- 153

with

SOLUTION

Step 1

Find sin a and cos b.

Because cos a = Quadrant III, sin

---45 a =

and a is ---35, as

in

shown in the figure.

Because sin b = -- 153 Quadrant I, cos b =

and -- 1123,

b is in as shown

in the figure.

y

y

4

a

x

52 - 42 = 3

5

13

b

5

x

132 - 52 = 12

Step 2 Use the difference formula for cosine to find cos(a - b).

cos(a - b) = cos a cos b + sin a sin b Difference formula for cosine

( ) ( )( ) = ---54 -- 1132 + ---53 -- 153

Evaluate.

= --- 6653 The value of cos(a - b) is --- 6635.

Simplify.

Simplifying an Expression

Simplify the expression cos(x + ).

SOLUTION cos(x + ) = cos x cos - sin x sin = (cos x)(-1) - (sin x)(0) = -cos x

Sum formula for cosine Evaluate. Simplify.

Monitoring Progress

Help in English and Spanish at

Find the exact value of the expression.

1. sin 105?

2. cos 15?

3. tan -- 512

4. cos -- 12

5.

Find with

sin(a - b) given < b < -- 32.

that

sin

a

=

-- 187

with

0

<

a

<

--2

and

cos

b

=

--- 2254

Simplify the expression.

6. sin(x + )

7. cos(x - 2)

8. tan(x - )

Section 9.8 Using Sum and Difference Formulas 521

Solving Equations and Rewriting Formulas

ANOTHER WAY

You can also solve the equation by using a graphing calculator. First, graph each side of the original equation. Then use the intersect feature to find the x-value(s) where the expressions are equal.

air

light

prism

Solving a Trigonometric Equation

( ) ( ) Solve sin x + --3 + sin x - --3 = 1 for 0 x < 2.

SOLUTION

( ) ( ) sin x + --3 + sin x - --3 = 1

sin x cos --3 + cos x sin --3 + sin x cos --3 - cos x sin --3 = 1 --12sin x + -- 2--3 cos x + --21sin x - -- 2--3 cos x = 1 sin x = 1

In the interval 0 x < 2, the solution is x = --2 .

Write equation. Use formulas. Evaluate. Simplify.

Rewriting a Real-Life Formula

The index of refraction of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. A triangular prism, like the one shown, can be used to measure the index of refraction using the formula

( ) For

n

= =

-- sin s--2in+--2 --2 . 60?, show that

the

formula

can

be

rewritten

as

n

=

-- 2--3

+

-- 2 1

cot

--2.

SOLUTION

( ) n

=

sin --2 + 30? -- sin-- --2

= -- sin --2 cos 3-- 0?si+n --2cos-- --2 sin 30?

( )( ) ( )( ) --

= -- sin --2 -- -- 23sin+--2 co-- s --2 --21

=

-- 2--3 sin --2 -- sin --2

+

-- --21scinos--2--2

=

--

-- 23

+

-- 1 2

cot

-- 2

Write

formula

with

-- 2

=

-- 620?

=

30?.

Sum formula for sine

Evaluate.

Write as separate fractions. Simplify.

Monitoring Progress

Help in English and Spanish at

( ) ( ) 9. Solve sin --4 - x - sin x + --4 = 1 for 0 x < 2.

522 Chapter 9 Trigonometric Ratios and Functions

9.8 Exercises

Dynamic Solutions available at

Vocabulary and Core Concept Check

1. COMPLETE THE SENTENCE Write the expression cos 130? cos 40? - sin 130? sin 40? as the cosine of an angle.

2. WRITING Explain how to evaluate tan 75? using either the sum or difference formula for tangent.

Monitoring Progress and Modeling with Mathematics

In Exercises 3?10, find the exact value of the expression. (See Example 1.)

3. tan(-15?)

4. tan 195?

5. sin -- 2132 7. cos 105? 9. tan -- 1172

6. sin(-165?)

8. cos -- 1112

( ) 10. sin --- 712

In Exercises 11?16, evaluate the expression given

that cos

--32 < b

a <

= --54

2.

with (See

0 < a < --2 and

Example 2.)

sin

b

=

---1175

with

11. sin(a + b)

12. sin(a - b)

13. cos(a - b)

14. cos(a + b)

15. tan(a + b)

16. tan(a - b)

In Exercises 17?22, simplify the expression. (See Example 3.)

17. tan(x + )

( ) 18. cos x - --2

19. cos(x + 2)

20. tan(x - 2)

( ) 21. sin x - -- 32

( ) 22. tan x + --2

ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in simplifying the expression.

23.

( ) tan x + --4 = -- 1t+antxa+n-- xttaann--4--4

= -- t1a+n xt+an1x

= 1

24.

( ) sin x - --4 = sin --4 cos x - cos --4 sin x

= --2-- 2 cos x - --2-- 2 sin x

= --2-- 2 (cos x - sin x)

25. What are the solutions of the equation 2 sin x - 1 = 0

for 0 x < 2?

A --3

B --6

C -- 23

D -- 56

26. What are the solutions of the equation tan x + 1 = 0 for 0 x < 2?

A --4

B -- 34

C -- 54

D -- 74

In Exercises 27? 32, solve the equation for 0 x < 2. (See Example 4.)

( ) 27. sin x + --2 = --12

( ) 28. tan x - --4 = 0

( ) ( ) 29. cos x + --6 - cos x - --6 = 1

( ) ( ) 30. sin x + --4 + sin x - --4 = 0

31. tan(x + ) - tan( - x) = 0

32. sin(x + ) + cos(x + ) = 0

33. USING EQUATIONS Derive the cofunction identity

( ) sin --2 - = cos using the difference formula

for sine.

Section 9.8 Using Sum and Difference Formulas 523

34. MAKING AN ARGUMENT Your friend claims it is possible to use the difference formula for tangent to

( ) derive the cofunction identity tan --2 - = cot . Is

your friend correct? Explain your reasoning.

35. MODELING WITH MATHEMATICS A photographer is at a height h taking aerial photographs with a 35-millimeter camera. The ratio of the image length WQ to the length NA of the actual object is given by the formula camera

Q

-- WNAQ = -- 35 tan( h-tat-- n) + 35 tan t

h W

t

N

A

where is the angle between the vertical line

perpendicular to the ground and the line from the

camera to point A and t is the tilt angle of the film.

When t = 45?, show that the formula can be rewritten

as -- WNAQ = -- h(1 +7-- 0tan ). (See Example 5.)

36. MODELING WITH MATHEMATICS When a wave travels through a taut string, the displacement y of each point on the string depends on the time t and the point's position x. The equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions. Suppose a standing wave can be modeled by the formula

( ) ( ) y = A cos -- 23t - -- 25x + A cos -- 23t + -- 25x .

When t = 1, show that the formula can be rewritten as y = -A cos -- 25 x .

37. MODELING WITH MATHEMATICS The busy signal on a touch-tone phone is a combination of two tones with frequencies of 480 hertz and 620 hertz. The individual tones can be modeled by the equations:

480 hertz: y1 = cos 960t

620 hertz: y2 = cos 1240t

The sound of the busy signal can be modeled by y1 + y2. Show that y1 + y2 = 2 cos 1100t cos 140t.

( (

38. HOW DO YOU SEE IT? Explain how to use the figure

( ) ( ) to solve the equation sin x + --4 - sin --4 - x = 0

for 0 x < 2.

( y

f(x) = sin

x

+

4

x

2

-1

( g(x) = sin

4

-

x

39. MATHEMATICAL CONNECTIONS The figure shows the

acute angle of intersection, 2 - 1, of two lines with slopes m1 and m2.

y

y = m1x + b1

y = m2x + b2

2 - 1

1

2

x

a. Use the difference formula for tangent to write an equation for tan (2 - 1) in terms of m1 and m2.

b. Use the equation from part (a) to find the acute

angle of intersection of the lines y = x - 1 and

() --

y =

1 -- --3 - 2

x + -- 24 -- --33 .

40. THOUGHT PROVOKING Rewrite each function. Justify your answers. a. Write sin 3x as a function of sin x. b. Write cos 3x as a function of cos x. c. Write tan 3x as a function of tan x.

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Solve the equation. Check your solution(s). (Section 7.5)

41. 1 - -- x -9 2 = ---27

42. -- 1x2 + --43 = --8x

43. -- 2xx+-13 = -- x21-0 1 + 5

524 Chapter 9 Trigonometric Ratios and Functions

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