Question

write the expression as a sum or difference of trigonometric functions.
2 sin 51° cos 105°

2 sin 51° cos 105° = \square
(do not include the degree symbol in your answer.)

Explanation:

Step1: Recall the product-to-sum formula

The product - to - sum formula for \(2\sin A\cos B\) is \(\sin(A + B)+\sin(A - B)\). Here, \(A = 51\) and \(B=105\).

Step2: Calculate \(A + B\) and \(A - B\)

First, calculate \(A + B\): \(A + B=51 + 105=156\)
Then, calculate \(A - B\): \(A - B=51-105=- 54\)
We know that \(\sin(-x)=-\sin x\), so \(\sin(51 - 105)=\sin(-54)=-\sin54\)
And \(\sin(51 + 105)=\sin156\)
So \(2\sin51\cos105=\sin(51 + 105)+\sin(51 - 105)=\sin156-\sin54\) (since \(\sin(-54)=-\sin54\))

Answer:

\(\sin156-\sin54\)