Find each exact function value. See Example 2.
cos 7π/4

Verified step by step guidance

1. Recognize that the angle given is \( \frac{7\pi}{4} \), which is in radians. This angle is located in the fourth quadrant of the unit circle because \( \frac{7\pi}{4} \) is between \( \frac{3\pi}{2} \) and \( 2\pi \).
2. Recall that the cosine function corresponds to the x-coordinate of a point on the unit circle at the given angle.
3. Find the reference angle for \( \frac{7\pi}{4} \) by subtracting it from \( 2\pi \): \( 2\pi - \frac{7\pi}{4} = \frac{\pi}{4} \).
4. Use the known cosine value for the reference angle \( \frac{\pi}{4} \), which is \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).
5. Since cosine is positive in the fourth quadrant, the value of \( \cos \frac{7\pi}{4} \) is the same as \( \cos \frac{\pi}{4} \), so \( \cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2} \).

Key Concepts

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to arc lengths on this circle. Understanding how to locate an angle like 7π/4 radians on the unit circle is essential for finding exact trigonometric values.

Cosine Function on the Unit Circle

The cosine of an angle corresponds to the x-coordinate of the point on the unit circle at that angle. By identifying the position of 7π/4 on the unit circle, you can determine the exact cosine value by reading the x-coordinate of that point.

Reference Angles and Quadrants

Reference angles help simplify finding trigonometric values by relating any angle to an acute angle in the first quadrant. Knowing the quadrant of 7π/4 (fourth quadrant) allows you to determine the sign of the cosine value, as cosine is positive in the fourth quadrant.