Question

Evaluate the function f(x)=cos(−x) at x=−
3
4π
​
.

Answer 1

To evaluate the function \( f(x) = \cos(-x) \) at \( x = -\frac{3\pi}{4} \), we first substitute \( x \) with \(-\frac{3\pi}{4}\):

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

Next, we calculate \( \cos\left(\frac{3\pi}{4}\right) \). The angle \( \frac{3\pi}{4} \) is in the second quadrant, where the cosine is negative. Specifically:

\[
\frac{3\pi}{4} = \pi - \frac{\pi}{4}
\]

We know that:

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

Since \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), we find:

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

Thus,

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

Therefore, the final answer is:

\[
\boxed{-\frac{\sqrt{2}}{2}}
\]