In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of


0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋.
4 2 4 4 2 4


a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
cos 3𝜋/4

Verified step by step guidance

1. Understand that the problem involves finding the value of \( \cos \frac{3\pi}{4} \) using the unit circle, where each point on the circle corresponds to an angle \( t \) and has coordinates \( (x, y) = (\cos t, \sin t) \).
2. Locate the angle \( \frac{3\pi}{4} \) on the unit circle. Since the circle is divided into eight equal arcs, each arc corresponds to an angle of \( \frac{2\pi}{8} = \frac{\pi}{4} \). The angle \( \frac{3\pi}{4} \) is the third division from 0, moving counterclockwise.
3. Identify the coordinates of the point on the unit circle at \( t = \frac{3\pi}{4} \). This point lies in the second quadrant, where the x-coordinate (cosine) is negative and the y-coordinate (sine) is positive.
4. Recall the exact values for cosine and sine at multiples of \( \frac{\pi}{4} \). Specifically, \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \) and \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).
5. Conclude that the value of \( \cos \frac{3\pi}{4} \) is the x-coordinate of the point on the unit circle at that angle, which is \( -\frac{\sqrt{2}}{2} \).

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, starting from the positive x-axis and moving counterclockwise. Understanding how angles correspond to points on the circle is essential for evaluating trigonometric functions like cosine and sine.

Coordinates on the Unit Circle

Each point on the unit circle corresponds to an angle t and has coordinates (x, y), where x = cos(t) and y = sin(t). These coordinates represent the cosine and sine values of the angle, respectively. Knowing how to read or find these coordinates allows direct evaluation of trigonometric functions.

Cosine Function and Its Geometric Interpretation

The cosine of an angle in the unit circle is the x-coordinate of the corresponding point. For angles like 3𝜋/4, which lie in the second quadrant, cosine values are negative. Recognizing the quadrant helps determine the sign and value of cosine without memorizing all values.