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Section 9.8 Using Sum and Difference Formulas 519
9.8 Using Sum and Difference Formulas
Essential QuestionEssential 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.
b
a
(cos b , sin b )
(cos a , sin a )
x
y (^) d
1
x
y
(1, 0)
1 a − b
d
(cos( a − b ), sin( a − b ))
b. Use the Distance Formula to write an expression for d in the fi rst 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(
π — 2
− a ) = cos a and cos(
π — 2
− a ) = sin a
and the fact that
cos[( π — 2 − a (^) ) + b ] = sin( a − b ) and sin( a + b ) = sin[ a − (− b )].
Communicate Your AnswerCommunicate 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).
CONSTRUCTING
VIABLE ARGUMENTS
To be proficient in math, you need to understand and use stated assumptions, defi nitions, and previously established results.
520 Chapter 9 Trigonometric Ratios and Functions
9.8 Lesson^ What You Will LearnWhat 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.
In general, sin( a + b ) ≠ sin a + sin b. Similar statements can be made for the other trigonometric functions of sums and differences.
Evaluating Trigonometric Expressions
Find the exact value of (a) sin 15° and (b) tan 7 π — 12
SOLUTION
a. sin 15° = sin(60° − 45 °) Substitute 60° − 45 ° for 15°. = sin 60° cos 45° − cos 60° sin 45° Difference formula for sine
=
— 3 — 2 (^
— 2 — 2 )^
— 2 (^
— 2 — 2 )^
Evaluate.
— 6 − √
— 2 — 4
Simplify.
The exact value of sin 15° is
— 6 − √
— 2 — 4
. Check this with a calculator.
b. tan 7 π — 12
= tan( π — 3
π — 4 )^
Substitute π — 3
π — 4
for^7 —^ π 12
tan π — 3
- tan π — 4 —— 1 − tan^ π— 3
tan^ π— 4
Sum formula for tangent
— 3 + 1 — 1 − √
— (^3) ⋅ 1
Evaluate.
— 3 Simplify.
The exact value of tan 7 π — 12
is − 2 − √
—
- Check this with a calculator.
Check sin(15˚) .
.
( (6)- (2))/
Check
tan( -3.
-3.
7π
-2- (3)
/12)
Previous ratio
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
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
tan( a + b ) = tan a + tan b —— 1 − tan a tan 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 ) = tan a − tan b —— 1 + tan a tan b
522 Chapter 9 Trigonometric Ratios and Functions
Solving Equations and Rewriting Formulas
Solving a Trigonometric Equation
Solve sin( x +^ π— 3 )^
= 1 for 0 ≤ x < 2 π.
SOLUTION
sin( x +^ π— 3 )^
= 1 Write equation.
sin x cos^ π— 3
cos x sin^ π— 3
sin x cos^ π— 3
− cos x sin—^ π 3
= 1 Use formulas.
1 — 2
sin x + √
— 3 — 2
cos x + (^1) — 2
sin x − √
— 3 — 2
cos x = 1 Evaluate.
sin x = 1 Simplify.
In the interval 0 ≤ x < 2 π, the solution is x =^ π— 2
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
n =
sin( θ — 2
α — 2 ) — sin^ θ— 2
For α = 60 °, show that the formula can be rewritten as n = √
— 3 — 2
cot^ θ— 2
SOLUTION
n =
sin( θ — 2
Write formula with—^ α 2
sin θ — 2
cos 30° + cos θ — 2
sin 30° ——— sin^ θ— 2
Sum formula for sine
( sin
θ — 2 ) (^
— 3 — 2 )^
— 2 ) ——— sin^ θ— 2
Evaluate.
— 3 — 2
sin θ — 2 — sin^ θ— 2
— 2
cos θ — 2 — sin—^ θ 2
Write as separate fractions.
— 3 — 2
+ —^1
cot^ θ— 2
Simplify.
Monitoring ProgressMonitoring Progress (^) Help in English and Spanish at BigIdeasMath.com
9. Solve sin(^ π—
− x ) − sin( x +^ π—
4 )^
= 1 for 0 ≤ x < 2 π.
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 fi nd the x -value(s) where the expressions are equal.
prism
air
light
θ
α
Section 9.8 Using Sum and Difference Formulas 523
9.8 Exercises Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–10, find the exact value of the expression. (See Example 1.)
3. tan(− 15 °) 4. tan 195° 5. sin^23 —^ π 12 6. sin(− 165 °) 7. cos 105° 8. cos^11 —^ π 12 9. tan^17 —^ π 12
10. sin( −^7 —^ π
In Exercises 11–16, evaluate the expression given
that cos a =
— 5 with 0 <^ a^ <
π — 2 and sin^ b^ =^ −
— 17 with 3 π — 2 <^ b^ < 2^ π.^ (See Example 2.)
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 + π) 18. cos( x −—^ π
19. cos( x + 2 π) 20. tan( x − 2 π)
21. sin( x − 3 —^ π
2 )^
22. tan( x +^ π—
ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in simplifying the expression.
23.
tan( x +
π
tan x + tan π — 4 —— 1 + tan x tan^ π— 4 = tan x + 1 — 1 + tan x
= 1
sin ( x −
π
— 4 ) =^ sin
π — 4 cos^ x^ −^ cos
π — 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 π?
A
π — 3
B
π — 6
C
2 π — 3
D
5 π — 6
26. What are the solutions of the equation tan x + 1 = 0 for 0 ≤ x < 2 π?
A
π — 4
B
3 π — 4
C
5 π — 4
D
7 π — 4
In Exercises 27– 32, solve the equation for 0 ≤ x < 2 π. (See Example 4.)
27. sin( x +
π —
2 )^
— 2
28. tan( x −
π —
4 )^
29. cos( x +
π —
6 )^
− cos( x −
π —
6 )^
30. sin( x +
π —
4 )^
+ sin( x −
π —
4 )^
31. tan( x + π) − tan( π − x ) = 0 32. sin( x + π) + cos( x + π) = 0 33. USING EQUATIONS Derive the cofunction identity
sin(
π — 2
− θ ) = cos θ using the difference formula
for sine.
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
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.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
