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9.2 Sum and Difference Identities

Formulas for cos(a ± b), sin(a ± b), and tan(a ± b), giving exact values at angles like 15° and quick cofunction proofs.

Algebra and Trigonometry 2e, 9.2 Sum and Difference Identities

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37 exercises

Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

Using the formula for the cosine of the difference of two angles, find the exact value of cos( 5π 4 − π 6 ). \cos \left(\frac{5 \pi}{4} - \frac{\pi}{6}\right) .

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9.2 Exercise 2

Find the exact value of cos( π 3 − π 4 ). \cos \left(\frac{\pi}{3} - \frac{\pi}{4}\right) .

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Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine

Find the exact value of cos(75°). \cos \left(75°\right) .

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9.2 Exercise 4

Find the exact value of cos(105°). \cos \left(105°\right) .

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Using Sum and Difference Identities to Evaluate the Difference of Angles

Use the sum and difference identities to evaluate the difference of the angles and show that part a equals part b.

  1. ⓐ sin(45°−30°) \sin \left(45° - 30°\right)
  2. ⓑ sin(135°−120°) \sin \left(135° - 120°\right)
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Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

Find the exact value of sin( cos −1 1 2 + sin −1 3 5 ). \sin \left(\cos^{−1} \frac{1}{2} + \sin^{−1} \frac{3}{5}\right) . Then check the answer with a graphing calculator.

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Finding the Exact Value of an Expression Involving Tangent

Find the exact value of tan( π 6 + π 4 ). \tan \left(\frac{\pi}{6} + \frac{\pi}{4}\right) .

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9.2 Exercise 8

Find the exact value of tan( 2π 3 + π 4 ). \tan \left(\frac{2 \pi}{3} + \frac{\pi}{4}\right) .

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Finding a Cofunction with the Same Value as the Given Expression

Write tan π 9 \tan \frac{\pi}{9} in terms of its cofunction.

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9.2 Exercise 11

Write sin π 7 \sin \frac{\pi}{7} in terms of its cofunction.

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Verifying an Identity Involving Sine

Verify the identity sin(α+β)+sin(α−β)=2sinαcosβ. \sin \left(\alpha + \beta\right) + \sin \left(\alpha - \beta\right) = 2 \sin \alpha \cos \beta .

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Verifying an Identity Involving Tangent

Verify the following identity.

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9.2 Exercise 14

Verify the identity: tan( π−θ )=−tanθ. \tan \left(\pi - \theta\right) = - \tan \theta .

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9.2 Exercise 17

Explain the basis for the cofunction identities and when they apply.

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9.2 Exercise 19

Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f(x)=sin(x) f \left(x\right) = \sin \left(x\right) and g(x)=cos(x). g \left(x\right) = \cos \left(x\right) . (Hint: 0−x=−x 0 - x = - x )

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9.2 Exercise 21

Find the exact value.

cos( π 12 ) \cos \left(\frac{\pi}{12}\right)

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9.2 Exercise 23

Find the exact value.

sin( 11π 12 ) \sin \left(\frac{11 \pi}{12}\right)

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9.2 Exercise 25

Find the exact value.

tan( 19π 12 ) \tan \left(\frac{19 \pi}{12}\right)

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9.2 Exercise 27

Rewrite in terms of sinx \sin x and cosx. \cos x .

sin( x− 3π 4 ) \sin \left(x - \frac{3 \pi}{4}\right)

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9.2 Exercise 29

Rewrite in terms of sinx \sin x and cosx. \cos x .

cos( x+ 2π 3 ) \cos \left(x + \frac{2 \pi}{3}\right)

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9.2 Exercise 31

Simplify the given expression.

sec( π 2 −θ ) \sec \left(\frac{\pi}{2} - \theta\right)

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9.2 Exercise 33

Simplify the given expression.

tan( π 2 −x ) \tan \left(\frac{\pi}{2} - x\right)

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9.2 Exercise 35

Simplify the given expression.

tan( 3 2 x )−tan( 7 5 x ) 1+tan( 3 2 x )tan( 7 5 x ) \frac{\tan \left(\frac{3}{2} x\right) - \tan \left(\frac{7}{5} x\right)}{1 + \tan \left(\frac{3}{2} x\right) \tan \left(\frac{7}{5} x\right)}

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9.2 Exercise 37

Find the requested information.

Given that sina= 4 5 , \sin a = \frac{4}{5} , and cosb= 1 3 , \cos b = \frac{1}{3} , with a a and b b both in the interval [ 0, π 2 ), \left[0 , \frac{\pi}{2}\right) , find sin(a−b) \sin \left(a - b\right) and cos(a+b). \cos \left(a + b\right) .

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9.2 Exercise 39

Find the exact value of each expression.

cos( cos −1 ( 2 2 )+ sin −1 ( 3 2 ) ) \cos \left(\cos^{- 1} \left(\frac{\sqrt{2}}{2}\right) + \sin^{- 1} \left(\frac{\sqrt{3}}{2}\right)\right)

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9.2 Exercise 49

Use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x. 2 x = x + x . )

f( x )=sin( 4x )−sin( 3x )cosx,g( x )=sinxcos( 3x ) f \left(x\right) = \sin \left(4 x\right) - \sin \left(3 x\right) \cos x , g \left(x\right) = \sin x \cos \left(3 x\right)

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9.2 Exercise 51

Use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x. 2 x = x + x . )

f( x )=sin( 3x )cos( 6x ),g( x )=−sin( 3x )cos( 6x ) f \left(x\right) = \sin \left(3 x\right) \cos \left(6 x\right) , g \left(x\right) = - \sin \left(3 x\right) \cos \left(6 x\right)

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9.2 Exercise 53

Use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x. 2 x = x + x . )

f(x)=sin(2x),g(x)=2sinxcosx f \left(x\right) = \sin \left(2 x\right) , g \left(x\right) = 2 \sin x \cos x

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9.2 Exercise 55

Use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x. 2 x = x + x . )

f(θ)=tan(2θ),g(θ)= tanθ 1+ tan 2 θ f \left(\theta\right) = \tan \left(2 \theta\right) , g \left(\theta\right) = \frac{\tan \theta}{1 + \tan^{2} \theta}

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9.2 Exercise 57

Use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x. 2 x = x + x . )

f(x)=tan(−x),g(x)= tanx−tan(2x) 1−tanxtan(2x) f \left(x\right) = \tan \left(- x\right) , g \left(x\right) = \frac{\tan x - \tan \left(2 x\right)}{1 - \tan x \tan \left(2 x\right)}

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9.2 Exercise 59

Find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

sin(195°) \sin \left(195°\right)

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9.2 Exercise 61

Find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

cos(345°) \cos \left(345°\right)

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9.2 Exercise 63

Prove the identities provided.

tan(x+ π 4 )= tanx+1 1−tanx \tan \left(x + \frac{\pi}{4}\right) = \frac{\tan x + 1}{1 - \tan x}

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9.2 Exercise 65

Prove the identities provided.

cos(a+b) cosacosb =1−tanatanb \frac{\cos \left(a + b\right)}{\cos a \cos b} = 1 - \tan a \tan b

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9.2 Exercise 67

Prove the identities provided.

cos(x+h)−cosx h =cosx cosh−1 h −sinx sinh h \frac{\cos \left(x + h\right) - \cos x}{h} = \cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h}

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9.2 Exercise 69

Prove or disprove the statements.

tan(u−v)= tanu−tanv 1+tanutanv \tan \left(u - v\right) = \frac{\tan u - \tan v}{1 + \tan u \tan v}

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9.2 Exercise 71

Prove or disprove the statements.

If α, β, \alpha , \beta , and γ \gamma are angles in the same triangle, then prove or disprove sin( α+β )=sinγ. \sin \left(\alpha + \beta\right) = \sin \gamma .

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.