Question

Rewrite the sine function f(x) = -2sin(4x - pi) + 3 as a cosine function

Answer 1

To rewrite the sine function \( f(x) = -2\sin(4x - \pi) + 3 \) as a cosine function, we can use the identity that relates sine and cosine:

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

In our case, let \( \theta = 4x - \pi \). We can rewrite \( \sin(4x - \pi) \) as:

\[
\sin(4x - \pi) = \cos\left((4x - \pi) - \frac{\pi}{2}\right) = \cos\left(4x - \frac{3\pi}{2}\right)
\]

Now we can substitute this expression back into our function:

\[
f(x) = -2\sin(4x - \pi) + 3
\]
\[
= -2\cos\left(4x - \frac{3\pi}{2}\right) + 3
\]

Combining the constant outside the cosine gives us:

\[
f(x) = 3 - 2\cos\left(4x - \frac{3\pi}{2}\right)
\]

Thus, the sine function \( f(x) = -2\sin(4x - \pi) + 3 \) can be rewritten as:

\[
f(x) = 3 - 2\cos\left(4x - \frac{3\pi}{2}\right)
\]