In Exercises 1–6, use the figures to find the exact value of each trigonometric function.
tan 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 tangent function definition: \( \tan(\beta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{28}{45} \).
4. Apply the double angle formula for tangent: \( \tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)} \).
5. Substitute \( \tan(\beta) = \frac{28}{45} \) into the double angle formula to find \( \tan(2\beta) \).

Key Concepts

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions include sine, cosine, and tangent, which are defined as ratios of the sides of a right triangle. For example, tangent (tan) of an angle is the ratio of the opposite side to the adjacent side. Understanding these functions is essential for solving problems involving angles and distances in triangles.

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For instance, the formula for tangent states that tan(2θ) = 2tan(θ) / (1 - tan²(θ)). These formulas are useful for simplifying expressions and solving equations involving angles that are multiples of a given angle, which is relevant for finding tan(2θ) in the given problem.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is crucial for determining the lengths of sides when only some dimensions are known, and it can be used to find the sine, cosine, and tangent values necessary for solving trigonometric problems.