Question

Write the function f(x) = -2sin(x-pi)+3 as a cosine

Answer 1

To rewrite the function \( f(x) = -2\sin(x - \pi) + 3 \) in terms of cosine, we can use the sine-to-cosine transformation. Recall that:

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

Thus, we can express \(\sin(x - \pi)\) as:

\[
\sin(x - \pi) = -\sin(x)
\]

This means:

\[
-2\sin(x - \pi) = 2\sin(x)
\]

Now, substituting \(\sin(x - \pi)\) into the original function, we have:

\[
f(x) = -2(-\sin(x)) + 3 = 2\sin(x) + 3
\]

Next, we can rewrite \(\sin(x)\) in terms of cosine:

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

Thus, we can write:

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

Therefore, in terms of cosine, the function is:

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

Thus, the final expression in terms of cosine is:

\[
f(x) = 2\cos(x) + 3
\]