Question

rewrite the sine function: f(x) = sin(2(x+3pi/2)) as a cosine

Answer 1

To rewrite the sine function \( f(x) = \sin\left(2\left(x + \frac{3\pi}{2}\right)\right) \) as a cosine function, we can use the identity:

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

First, let's simplify the argument of the sine function:

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

Now, we can express the sine function in terms of cosine:

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

Next, we simplify the argument of the cosine:

\[
(2x + 3\pi) - \frac{\pi}{2} = 2x + 3\pi - \frac{\pi}{2}
\]

To combine \( 3\pi \) and \( -\frac{\pi}{2} \):

\[
3\pi - \frac{\pi}{2} = \frac{6\pi}{2} - \frac{\pi}{2} = \frac{5\pi}{2}
\]

So we can write:

\[
f(x) = \cos\left(2x + \frac{5\pi}{2}\right)
\]

Thus, the rewritten form of the sine function as a cosine is:

\[
f(x) = \cos\left(2x + \frac{5\pi}{2}\right)
\]