To find a standard angle coterminal with [tex]\(\theta = -\frac{5\pi}{4}\)[/tex] and to evaluate [tex]\(\cos \theta\)[/tex], follow these steps:
1. Determine a standard coterminal angle for [tex]\(\theta = -\frac{5\pi}{4}\)[/tex]:
- A standard angle lies within [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex].
- To find a coterminal angle, you can add [tex]\(2\pi\)[/tex] (a full circle) to [tex]\(\theta\)[/tex] until the result is within the range of [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex].
Here, add [tex]\(2\pi\)[/tex] to [tex]\(-\frac{5\pi}{4}\)[/tex]:
[tex]\[
-\frac{5\pi}{4} + 2\pi = -\frac{5\pi}{4} + \frac{8\pi}{4} = \frac{3\pi}{4}
\][/tex]
Note that [tex]\(\frac{3\pi}{4}\)[/tex] is indeed within the interval [tex]\([0, 2\pi)\)[/tex], so it is a coterminal angle.
2. Evaluate [tex]\(\cos \theta\)[/tex] at the coterminal angle [tex]\(\frac{3\pi}{4}\)[/tex]:
- Recall the coordinates on the unit circle for angles in standard position.
- The angle [tex]\(\frac{3\pi}{4}\)[/tex] lies in the second quadrant where the angle from the positive x-axis to the terminal side is [tex]\(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\)[/tex].
In the second quadrant, the cosine value is negative, and for an angle of [tex]\(\frac{\pi}{4}\)[/tex], the cosine is [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. Therefore:
[tex]\[
\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
To summarize:
- The coterminal angle within the standard range for [tex]\(\theta = -\frac{5\pi}{4}\)[/tex] is [tex]\(\frac{3\pi}{4}\)[/tex].
- The cosine of [tex]\(\theta\)[/tex] at that angle is [tex]\(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex].
Thus, the final results are:
[tex]\[
\text{Coterminal Angle: } 2.356194490192345 \text{ (which is approximately } \frac{3\pi}{4}\text{)}
\][/tex]
[tex]\[
\cos \theta = -0.7071067811865475 \text{ (which is } -\frac{\sqrt{2}}{2} \text{ approximately)}
\][/tex]
