Find each exact function value. See Example 3.
sin 5π/6

Verified step by step guidance

1. Recognize that \( \frac{5\pi}{6} \) is an angle in radians.
2. Convert \( \frac{5\pi}{6} \) to degrees if needed: \( \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ \).
3. Identify the reference angle for \( 150^\circ \), which is \( 180^\circ - 150^\circ = 30^\circ \).
4. Recall that \( \sin(\theta) = \sin(180^\circ - \theta) \), so \( \sin(150^\circ) = \sin(30^\circ) \).
5. Use the known value \( \sin(30^\circ) = \frac{1}{2} \) to find \( \sin(150^\circ) \).

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 fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. Each point on the unit circle corresponds to an angle, allowing us to determine the exact values of these functions for various angles, including those expressed in radians.

Sine Function

The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. On the unit circle, it corresponds to the y-coordinate of a point at a given angle θ. Understanding the sine function is crucial for finding exact values, especially for common angles like π/6, π/4, and π/3.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help in determining the sine, cosine, and tangent values for angles greater than 90 degrees or less than 0 degrees. For example, the reference angle for 5π/6 is π/6, which allows us to find sin(5π/6) by using the known value of sin(π/6) and considering the sign based on the quadrant.