In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use 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.
cos 2𝜋/3

Verified step by step guidance

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

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 in trigonometry to define the sine, cosine, and tangent functions based on the coordinates of points on the circle. Each angle corresponds to a point on the circle, where the x-coordinate represents the cosine value and the y-coordinate represents the sine value.

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, these functions can be derived from the coordinates of points on the circle. For example, the cosine of an angle is the x-coordinate of the corresponding point, while the sine is the y-coordinate.

Angle Measurement

Angles in trigonometry can be measured in degrees or radians. The unit circle is typically used with radian measure, where a full rotation (360 degrees) corresponds to 2π radians. Understanding how to convert between degrees and radians is essential for accurately determining the trigonometric values associated with specific angles.