In Exercises 47–54, use the figures to find the exact value of each trigonometric function.θ θ2 sin ------- cos -------2 2

Verified step by step guidance

1. Identify the trigonometric identity for \( \sin(2\theta) \) and \( \cos(2\theta) \).
2. Recall that \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) and \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).
3. Substitute the given values of \( \sin(\theta) \) and \( \cos(\theta) \) from the figure into the identities.
4. Calculate \( \sin(2\theta) \) using the identity \( 2\sin(\theta)\cos(\theta) \).
5. Calculate \( \cos(2\theta) \) using the identity \( \cos^2(\theta) - \sin^2(\theta) \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), relate the angles of a triangle to the ratios of its sides. For a given angle θ in a right triangle, sin(θ) is the ratio of the length of the opposite side to the hypotenuse, while cos(θ) is the ratio of the adjacent side to the hypotenuse. Understanding these functions is essential for solving problems involving angles and lengths.

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles (2θ) in terms of single angles (θ). For example, sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) - sin²(θ). These formulas are crucial for simplifying expressions and calculating exact values of trigonometric functions when dealing with angles that are multiples of a given angle.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sin, cos, and other functions at key angles, such as 0°, 30°, 45°, 60°, and 90°. These values can be derived from the unit circle or special triangles. Knowing these exact values allows for quick calculations and is fundamental in solving trigonometric equations and problems.