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 $\sin (\theta )=\cos (\theta -\frac{\pi }{2})$, 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 $\sin (x-\pi )=-\cos (x)$.