In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.6 3 2 3 6 6 3 2 3 6Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.cos 3𝜋/2

Verified step by step guidance

1. Identify the angle \( \frac{3\pi}{2} \) on the unit circle.
2. Locate the corresponding point on the unit circle for \( \frac{3\pi}{2} \).
3. Observe that the point at \( \frac{3\pi}{2} \) is \((0, -1)\).
4. Recall that the cosine of an angle in the unit circle is the x-coordinate of the corresponding point.
5. Conclude that \( \cos \frac{3\pi}{2} \) is the x-coordinate of the point \((0, -1)\).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is used to define trigonometric functions for all real numbers. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, allowing for easy calculation of these functions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. On the unit circle, the cosine of an angle corresponds to the x-coordinate, while the sine corresponds to the y-coordinate of the point on the circle. Understanding these functions is essential for solving problems involving angles and their relationships.

Radians

Radians are a unit of angular measure used in mathematics, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. The unit circle divides the circle into radians, making it easier to work with angles in trigonometric functions. For example, 3π/2 radians corresponds to 270 degrees, which is a key angle in trigonometry.