Question
\cos \left(\frac{7\pi }{4}\right)
Answers
GPT-4o mini
To find \(\cos \left(\frac{7\pi}{4}\right)\), we can first note the angle \(\frac{7\pi}{4}\) is in radians. To understand its position on the unit circle, we can convert it to degrees:
\[
\frac{7\pi}{4} \times \frac{180^\circ}{\pi} = \frac{7 \times 180^\circ}{4} = \frac{1260^\circ}{4} = 315^\circ.
\]
The angle \(315^\circ\) is located in the fourth quadrant of the unit circle. The reference angle for \(315^\circ\) is:
\[
360^\circ - 315^\circ = 45^\circ.
\]
Now, the cosine of an angle in the fourth quadrant is positive. Thus:
\[
\cos \left(\frac{7\pi}{4}\right) = \cos(45^\circ).
\]
We know that:
\[
\cos(45^\circ) = \frac{\sqrt{2}}{2}.
\]
Therefore:
\[
\cos \left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}.
\]
Thus, the final answer is:
\[
\cos \left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}.
\]
\[
\frac{7\pi}{4} \times \frac{180^\circ}{\pi} = \frac{7 \times 180^\circ}{4} = \frac{1260^\circ}{4} = 315^\circ.
\]
The angle \(315^\circ\) is located in the fourth quadrant of the unit circle. The reference angle for \(315^\circ\) is:
\[
360^\circ - 315^\circ = 45^\circ.
\]
Now, the cosine of an angle in the fourth quadrant is positive. Thus:
\[
\cos \left(\frac{7\pi}{4}\right) = \cos(45^\circ).
\]
We know that:
\[
\cos(45^\circ) = \frac{\sqrt{2}}{2}.
\]
Therefore:
\[
\cos \left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}.
\]
Thus, the final answer is:
\[
\cos \left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}.
\]