Question

Rewrite the sine function f(x)=−2sin(x−π)+3 as a cosine function.
explain how you arrived at your answer.

Answer 1

To rewrite the sine function \( f(x) = -2\sin(x - \pi) + 3 \) as a cosine function, we can use the identity that relates sine and cosine. Specifically, the sine function can be expressed in terms of cosine as follows:

\[
\sin(x) = \cos\left(x - \frac{\pi}{2}\right)
\]

We also have the property that \(\sin(x - \pi) = -\sin(x)\). This will be helpful in rewriting our function.

1. **Rewrite the sine function**:
\[
\sin(x - \pi) = -\sin(x)
\]

2. **Substitute this into the original function**:
\[
f(x) = -2\sin(x - \pi) + 3 = -2(-\sin(x)) + 3 = 2\sin(x) + 3
\]

3. **Now express \(\sin(x)\) in terms of cosine**:
Using the identity mentioned above:
\[
\sin(x) = \cos\left(x - \frac{\pi}{2}\right)
\]
Thus,
\[
f(x) = 2\cos\left(x - \frac{\pi}{2}\right) + 3
\]

So, the function \( f(x) = -2\sin(x - \pi) + 3 \) can be rewritten as:

\[
f(x) = 2\cos\left(x - \frac{\pi}{2}\right) + 3
\]

### Summary:
1. We used the identity to relate sine to cosine.
2. The original sine function was first manipulated by noting \(\sin(x - \pi) = -\sin(x)\).
3. We expressed \( \sin(x) \) as \( \cos\left(x - \frac{\pi}{2}\right) \) to achieve the final cosine form. Thus, the rewritten function is \( f(x) = 2\cos\left(x - \frac{\pi}{2}\right) + 3 \).