Question

Rewrite the sine function f(x) = -2sin(x - pi) +3 as a cosine function. Explain how you got your answer

Answer 1

To rewrite the sine function \( f(x) = -2\sin(x - \pi) + 3 \) as a cosine function, we can use the relationship between sine and cosine.

Recall the identity:
\[
\sin(x - \pi) = -\cos(x)
\]
This is because sine and cosine are phase-shifted by \( \frac{\pi}{2} \) (or \( 90^\circ \)). Specifically, given that \(\sin(\theta) = \cos\left(\theta - \frac{\pi}{2}\right)\), we can express sine in terms of cosine.

Thus, we can rewrite the sine term in the function as follows:
1. Substitute \( \sin(x - \pi) \) with \(-\cos(x)\):
\[
f(x) = -2(-\cos(x)) + 3 = 2\cos(x) + 3
\]

So, the function in terms of cosine becomes:
\[
f(x) = 2\cos(x) + 3
\]

### Summary:
The sine function \( f(x) = -2\sin(x - \pi) + 3 \) is rewritten as \( f(x) = 2\cos(x) + 3 \) by using the identity that relates sine and cosine, specifically recognizing that \( \sin(x - \pi) = -\cos(x) \).