alleryStart learning →Which expression equals sin a cos b?
A(1/2)[sin(a + b) - sin(a - b)]
B(1/2)[cos(a - b) + cos(a + b)]
C(1/2)[sin(a + b) + sin(a - b)]
D(1/2)[cos(a - b) - cos(a + b)]
Answer & Solution
Correct answer: C. (1/2)[sin(a + b) + sin(a - b)]
1. Add sin(a + b) = sin a cos b + cos a sin b to sin(a - b) = sin a cos b - cos a sin b.
2. The cos a sin b terms cancel, leaving sin(a + b) + sin(a - b) = 2 sin a cos b.
3. Dividing by 2 gives sin a cos b = (1/2)[sin(a + b) + sin(a - b)].
4. The option with a minus sign between the sines equals cos a sin b, which reverses the roles of the two angles.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.4 Sum-to-Product and Product-to-Sum Formulas_
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