In Exercises 1–6, use the figures to find the exact value of each trigonometric function.
cos 2θ

Verified step by step guidance

1. Identify the given right triangle with sides 28, 45, and hypotenuse 53.
2. Recognize that the angle \( \beta \) is opposite the side of length 28 and adjacent to the side of length 45.
3. Use the cosine function for angle \( \beta \): \( \cos \beta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{45}{53} \).
4. Apply the double angle formula for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \).
5. Substitute \( \cos \beta \) into the double angle formula to find \( \cos 2\beta \).

Key Concepts

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. In a right triangle, the primary ratios are sine (sin), cosine (cos), and tangent (tan), defined as sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. These ratios are fundamental for solving problems involving right triangles and are essential for finding the values of trigonometric functions.

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For cosine, the formula is cos(2θ) = cos²(θ) - sin²(θ) or alternatively cos(2θ) = 2cos²(θ) - 1. Understanding these formulas is crucial for simplifying expressions and solving trigonometric equations involving double angles.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b), expressed as a² + b² = c². This theorem is essential for determining the lengths of sides in right triangles and is often used to derive trigonometric ratios, providing a foundation for further trigonometric analysis.