Question

Evaluate the function f(x)=cos(-x) at x=4π/3.

Answer 1

To evaluate the function \( f(x) = \cos(-x) \) at \( x = \frac{4\pi}{3} \), we proceed as follows:

First, substitute \( x \) into the function:

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

Using the property of cosine that \( \cos(-\theta) = \cos(\theta) \), we get:

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

Next, we need to find \( \cos\left(\frac{4\pi}{3}\right) \). The angle \( \frac{4\pi}{3} \) is located in the third quadrant, where the cosine function is negative.

To find the reference angle, we can subtract \( \pi \) (which is \( \frac{3\pi}{3} \)) from \( \frac{4\pi}{3} \):

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

Now we can evaluate \( \cos\left(\frac{4\pi}{3}\right) \):

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

Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \), we have:

\[
\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}
\]

Therefore, we conclude:

\[
f\left(\frac{4\pi}{3}\right) = \cos\left(-\frac{4\pi}{3}\right) = -\frac{1}{2}
\]

Thus, the final result is:

\[
\boxed{-\frac{1}{2}}
\]