alleryStart learning →Express sin(4t) cos(2t) as a sum containing no products.
A(1/2)[sin(6t) - sin(2t)]
B(1/2)[cos(6t) + cos(2t)]
C(1/2)[sin(2t) - sin(6t)]
D(1/2)[sin(6t) + sin(2t)]
Answer & Solution
Correct answer: D. (1/2)[sin(6t) + sin(2t)]
1. Use sin a cos b = (1/2)[sin(a + b) + sin(a - b)] with a = 4t and b = 2t.
2. The sum of the angles is 4t + 2t = 6t.
3. The difference is 4t - 2t = 2t.
4. So the expression becomes (1/2)[sin(6t) + sin(2t)].
5. The option with a minus sign between the sines would be the formula for cos a sin b, which swaps which angle carries the sine.
_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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