Certainly! Let's find the value of [tex]\(\cos \left(-\frac{\pi}{4}\right)\)[/tex].
We start by recalling a crucial property of the cosine function: cosine is an even function. This means that for any angle [tex]\( x \)[/tex],
[tex]\[
\cos(-x) = \cos(x).
\][/tex]
Given the angle [tex]\(-\frac{\pi}{4}\)[/tex], we can use this property to simplify the expression:
[tex]\[
\cos \left(-\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right).
\][/tex]
Now, let's determine the value of [tex]\(\cos \left(\frac{\pi}{4}\right)\)[/tex]. The angle [tex]\(\frac{\pi}{4}\)[/tex] is one of the common angles in trigonometry, and its cosine value is well-known. The value of [tex]\(\cos \left(\frac{\pi}{4}\right)\)[/tex] is:
[tex]\[
\cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}.
\][/tex]
To make the answer more presentable, we often rationalize the denominator. Therefore:
[tex]\[
\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.
\][/tex]
Thus, the value of [tex]\(\cos \left(\frac{\pi}{4}\right)\)[/tex] simplifies to:
[tex]\[
\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.
\][/tex]
However, the numerical value of [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is approximately [tex]\(0.7071067811865476\)[/tex].
Therefore, the value of [tex]\(\cos \left(-\frac{\pi}{4}\right)\)[/tex] is:
[tex]\[
0.7071067811865476.
\][/tex]
That's the detailed solution for finding the value of [tex]\(\cos \left(-\frac{\pi}{4}\right)\)[/tex].
