Question
\cos \left(\frac{5\pi }{4}\right)
Answers
GPT-4o mini
To find \(\cos\left(\frac{5\pi}{4}\right)\), we first need to determine the angle in reference to the unit circle.
The angle \(\frac{5\pi}{4}\) radians corresponds to \(225^\circ\) since:
\[
\frac{5\pi}{4} \times \frac{180^\circ}{\pi} = 225^\circ
\]
This angle is located in the third quadrant of the unit circle, where the cosine value is negative.
The reference angle for \(\frac{5\pi}{4}\) is given by:
\[
\frac{5\pi}{4} - \pi = \frac{5\pi}{4} - \frac{4\pi}{4} = \frac{\pi}{4}
\]
The cosine of the reference angle \(\frac{\pi}{4}\) is:
\[
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
Since we are in the third quadrant, where cosine is negative, we have:
\[
\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
Thus, the answer is:
\[
\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
The angle \(\frac{5\pi}{4}\) radians corresponds to \(225^\circ\) since:
\[
\frac{5\pi}{4} \times \frac{180^\circ}{\pi} = 225^\circ
\]
This angle is located in the third quadrant of the unit circle, where the cosine value is negative.
The reference angle for \(\frac{5\pi}{4}\) is given by:
\[
\frac{5\pi}{4} - \pi = \frac{5\pi}{4} - \frac{4\pi}{4} = \frac{\pi}{4}
\]
The cosine of the reference angle \(\frac{\pi}{4}\) is:
\[
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
Since we are in the third quadrant, where cosine is negative, we have:
\[
\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
Thus, the answer is:
\[
\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]