Let's solve the problem step by step.
### Step 1: Find a Coterminal Angle
To find a standard angle coterminal with [tex]\(\theta = -\frac{5\pi}{4}\)[/tex], we need to bring this angle within the range of [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex]. Coterminal angles are angles that differ by an integer multiple of [tex]\(2\pi\)[/tex]. We can find a coterminal angle in the desired range by adding [tex]\(2\pi\)[/tex] to [tex]\(\theta\)[/tex] repeatedly until the resulting angle is within the range of 0 to [tex]\(2\pi\)[/tex].
Starting with:
[tex]\[
\theta = -\frac{5\pi}{4}
\][/tex]
Add [tex]\(2\pi\)[/tex] to [tex]\(\theta\)[/tex]:
[tex]\[
\theta + 2\pi = -\frac{5\pi}{4} + 2\pi = -\frac{5\pi}{4} + \frac{8\pi}{4} = \frac{3\pi}{4}
\][/tex]
So the coterminal angle in the range from [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex] is:
[tex]\[
\coterminal\ angle = \frac{3\pi}{4}
\][/tex]
### Step 2: Evaluate [tex]\(\cos \theta\)[/tex]
Now, we use the unit circle to evaluate [tex]\(\cos \theta\)[/tex] for [tex]\(\theta = \frac{3\pi}{4}\)[/tex].
The angle [tex]\(\frac{3\pi}{4}\)[/tex] is positioned in the second quadrant of the unit circle, where the cosine function (which corresponds to the x-coordinate of the unit circle) is negative.
From our standard knowledge of the unit circle:
[tex]\[
\cos \left( \frac{3\pi}{4} \right) = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}
\][/tex]
Thus, the cosine of the original angle [tex]\(\theta = -\frac{5\pi}{4}\)[/tex] is:
[tex]\[
\cos\left( -\frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2}
\][/tex]
### Conclusion
To summarize, we found the standard angle coterminal with [tex]\(\theta = -\frac{5\pi}{4}\)[/tex] to be [tex]\(\frac{3\pi}{4}\)[/tex]. Using the unit circle, we evaluated:
[tex]\[
\cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2}
\][/tex]
Hence:
[tex]\[
\coterminal\ angle = 2.356194490192345 \quad (\text{approximately}\ \frac{3\pi}{4}), \quad \text{and} \quad \cos \theta = -0.7071067811865475 \quad (\text{approximately}\ -\frac{\sqrt{2}}{2})
\][/tex]
