In Exercises 47–54, use the figures to find the exact value of each trigonometric function.θsin -------2

Verified step by step guidance

1. Identify the given angle \( \theta \) and the trigonometric function you need to find, which is \( \sin \frac{\theta}{2} \).
2. Recall the half-angle identity for sine: \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \).
3. Determine the value of \( \cos \theta \) using the given figure or information.
4. Substitute the value of \( \cos \theta \) into the half-angle identity formula.
5. Decide the sign (positive or negative) of \( \sin \frac{\theta}{2} \) based on the quadrant in which \( \frac{\theta}{2} \) lies.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. The sine function, for example, is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding these functions is essential for solving problems involving angles and distances.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric representation of the trigonometric functions, where the x-coordinate corresponds to the cosine of an angle and the y-coordinate corresponds to the sine. Familiarity with the unit circle helps in determining the exact values of trigonometric functions for common angles.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often expressed in terms of square roots. For example, sin(30°) equals 1/2, and cos(45°) equals √2/2. Knowing these exact values is crucial for solving trigonometric equations and simplifying expressions without relying on calculators.