Question

use the product - to - sum identities to rewrite the following expression as a sum or difference.
5sin(95°)cos(140°)

Explanation:

Step1: Recall product - to - sum formula

The product - to - sum formula for $\sin A\cos B$ is $\sin A\cos B=\frac{1}{2}[\sin(A + B)+\sin(A - B)]$. Here $A = 95^{\circ}$ and $B=140^{\circ}$.

Step2: Apply the formula to the given expression

We have $5\sin(95^{\circ})\cos(140^{\circ})=5\times\frac{1}{2}[\sin(95^{\circ}+ 140^{\circ})+\sin(95^{\circ}-140^{\circ})]$.

Step3: Simplify the angles inside the sine functions

$95^{\circ}+140^{\circ}=235^{\circ}$ and $95^{\circ}-140^{\circ}=- 45^{\circ}$. Since $\sin(-\theta)=-\sin\theta$, we get $5\times\frac{1}{2}[\sin(235^{\circ})+\sin(-45^{\circ})]=\frac{5}{2}[\sin(235^{\circ})-\sin(45^{\circ})]$.

Answer:

$\frac{5}{2}[\sin(235^{\circ})-\sin(45^{\circ})]$