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 given: \(t = \frac{2\pi}{3}\).
2. Locate the point on the unit circle corresponding to \(t = \frac{2\pi}{3}\). From the image, this point is \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).
3. Recall that on the unit circle, the coordinates \((x, y)\) correspond to \((\cos t, \sin t)\) respectively.
4. Since we need to find \(\cos \frac{2\pi}{3}\), take the x-coordinate of the point, which is \(-\frac{1}{2}\).
5. Therefore, \(\cos \frac{2\pi}{3} = -\frac{1}{2}\).

Key Concepts

Unit Circle and Coordinates

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t measured in radians from the positive x-axis. The coordinates (x, y) of each point represent the cosine and sine of the angle t, respectively.

Trigonometric Functions on the Unit Circle

Cosine and sine functions can be directly found from the x and y coordinates of points on the unit circle. For an angle t, cos(t) is the x-coordinate and sin(t) is the y-coordinate of the corresponding point on the circle.

Special Angles and Their Values

Certain angles, such as multiples of π/6 and π/3, have well-known exact trigonometric values. These values are often expressed in terms of square roots and fractions, allowing precise evaluation of trigonometric functions without a calculator.