Give the exact value of each expression. See Example 5.sin 30°

Verified step by step guidance

1. Identify the angle given in the problem, which is 30°.
2. Recall the exact trigonometric value for \( \sin 30^\circ \).
3. Use the fact that \( \sin 30^\circ \) is a well-known angle in trigonometry.
4. Remember that \( \sin 30^\circ \) corresponds to the ratio of the opposite side to the hypotenuse in a right triangle.
5. Conclude that \( \sin 30^\circ \) is equal to \( \frac{1}{2} \).

Key Concepts

Trigonometric Ratios

Trigonometric ratios are relationships between the angles and sides of a right triangle. The primary ratios include sine, cosine, and tangent, which are defined as the ratios of specific sides of the triangle. For example, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Understanding these ratios is essential for solving problems involving angles and lengths in trigonometry.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it allows for the definition of trigonometric functions for all angles, not just those in right triangles. The coordinates of points on the unit circle correspond to the cosine and sine values of the angles formed with the positive x-axis. This concept is crucial for finding exact values of trigonometric functions like sin 30°.

Special Angles

Special angles refer to specific angles in trigonometry that have known sine, cosine, and tangent values. Common special angles include 0°, 30°, 45°, 60°, and 90°. For instance, sin 30° is a special angle with a known value of 1/2. Familiarity with these angles allows for quick calculations and is essential for solving various trigonometric problems efficiently.