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

검증된 단계별 안내

Identify the given right triangle with sides: opposite = 11, adjacent = 60, hypotenuse = 61.

Use the tangent function for angle \( \alpha \): \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{11}{60} \).

Apply the double angle formula for tangent: \( \tan 2\alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} \).

Substitute \( \tan \alpha = \frac{11}{60} \) into the double angle formula.

Simplify the expression to find \( \tan 2\alpha \).

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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 is the ratio of the opposite side to the adjacent side. Understanding these functions is essential for solving problems involving angles and side lengths in triangles.

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For tangent, the formula is tan(2α) = 2tan(α) / (1 - tan²(α)). These formulas are useful for simplifying expressions and solving trigonometric equations, especially when dealing with angles that are multiples of a given angle.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is fundamental in trigonometry as it allows for the calculation of side lengths when angles are known, and vice versa. In the given triangle, it can be used to verify the relationships between the sides and to find missing values.

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