alleryStart learning →Using the difference formula, find the exact value of cos(5 pi / 4 - pi / 6).
A(sqrt(6) - sqrt(2)) / 4
B(sqrt(2) - sqrt(6)) / 4
C(-sqrt(6) - sqrt(2)) / 4
D(sqrt(6) + sqrt(2)) / 4
Answer & Solution
Correct answer: C. (-sqrt(6) - sqrt(2)) / 4
1. Write cos(a - b) = cos a cos b + sin a sin b with a = 5 pi / 4 and b = pi / 6.
2. The needed values are cos(5 pi / 4) = -sqrt(2)/2, cos(pi/6) = sqrt(3)/2, sin(5 pi / 4) = -sqrt(2)/2 and sin(pi/6) = 1/2.
3. The first product is (-sqrt(2)/2)(sqrt(3)/2) = -sqrt(6)/4.
4. The second product is (-sqrt(2)/2)(1/2) = -sqrt(2)/4.
5. Adding them gives -sqrt(6)/4 - sqrt(2)/4, that is (-sqrt(6) - sqrt(2)) / 4.
6. Both terms are negative because the angle 5 pi / 4 sits in the third quadrant, where sine and cosine are both negative.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.2 Sum and Difference Identities_
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