Express the product as a sum containing only sines or cosines.
-$$\sin \frac { \theta } { 2 } \cos \frac { 5 \theta } { 2 }$$
A) $\frac { 1 } { 4 } \sin \cos ( 5 \theta )$
B) $\frac { 1 } { 4 } [ \cos ( 6 \theta ) - \sin ( 4 \theta ) ]$
C) $\frac { 1 } { 2 } [ \cos ( 3 \theta ) + \sin ( 2 \theta ) ]$
D) $\frac { 1 } { 2 } [ \sin ( 3 \theta ) - \sin ( 2 \theta ) ]$

Question

Quizplus · Accepted Answer

The product-to-sum formulas can be used to express the product of sine and cosine as a sum. The relevant formula is $\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$. Applying this to $\sin \frac{\theta}{2} \cos \frac{5\theta}{2}$, we get $\frac{1}{2}[\sin(\frac{\theta}{2} + \frac{5\theta}{2}) + \sin(\frac{\theta}{2} - \frac{5\theta}{2})] = \frac{1}{2}[\sin(3\theta) - \sin(2\theta)]$, which matches option D.

Express the Product as a Sum Containing Only Sines or Cosines