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

Verified step by step guidance

1. insert step 1> Identify the given angle \( \theta \) and understand that you need to find \( \tan\left(\frac{\theta}{2}\right) \).
2. insert step 2> Recall the half-angle identity for tangent: \( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} \).
3. insert step 3> Determine \( \cos\theta \) from the given figure or information.
4. insert step 4> Substitute the value of \( \cos\theta \) into the half-angle identity.
5. insert step 5> Simplify the expression to find the exact value of \( \tan\left(\frac{\theta}{2}\right) \).

Key Concepts

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions include sine (sin), cosine (cos), and tangent (tan), which are defined as ratios of the sides of a right triangle. Understanding these functions is essential for solving problems involving angles and distances in trigonometry.

Tangent Function

The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ) / cos(θ). Knowing how to calculate and interpret the tangent function is crucial for finding exact values in trigonometric problems.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to specific values that can be determined without approximation, often using special angles like 0°, 30°, 45°, 60°, and 90°. These values are typically derived from the unit circle or special right triangles, and they are fundamental for solving trigonometric equations and problems accurately.