To determine the value of [tex]\( f\left(\frac{3 \pi}{8}\right) \)[/tex] for the function [tex]\( f(x) = \cos(2x) \)[/tex], we proceed as follows:
First, we need to find the value of [tex]\( 2x \)[/tex] when [tex]\( x = \frac{3\pi}{8} \)[/tex].
[tex]\[
x = \frac{3\pi}{8}
\][/tex]
Doubling [tex]\( x \)[/tex] we get:
[tex]\[
2x = 2 \left(\frac{3\pi}{8}\right) = \frac{6\pi}{8} = \frac{3\pi}{4}
\][/tex]
Next, we evaluate the cosine function at the angle [tex]\( \frac{3\pi}{4} \)[/tex]:
[tex]\[
\cos\left(\frac{3\pi}{4}\right)
\][/tex]
The angle [tex]\( \frac{3\pi}{4} \)[/tex] is in the second quadrant. In the second quadrant, the cosine function is negative. The reference angle for [tex]\( \frac{3\pi}{4} \)[/tex] is [tex]\( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \)[/tex].
From trigonometric values, we know that:
[tex]\[
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
Given that cosine is negative in the second quadrant:
[tex]\[
\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
Thus, the exact value of [tex]\( f\left( \frac{3\pi}{8} \right) \)[/tex] is:
[tex]\[
f\left(\frac{3\pi}{8}\right) = \cos\left(2 \times \frac{3\pi}{8}\right) = \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
But simplifying the result to make sure it matches our original numeric solution, given:
[tex]\[
f\left(\frac{3\pi}{8}\right) = -\frac{1}{\sqrt{2}}
\][/tex]
[tex]\[
- \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = - \frac{\sqrt{2}}{2}
\][/tex]
This value is [tex]\(-0.7071067811865475\)[/tex] as a numerical approximation, indicating the steps are correctly followed.
So, the exact value of [tex]\( f\left(\frac{3\pi}{8}\right) \)[/tex] is:
[tex]\[
f\left(\frac{3\pi}{8}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
