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

Verified step by step guidance

1. Identify the given right triangle with sides 60, 11, and hypotenuse 61.
2. Recognize that the angle \( \alpha \) is opposite the side of length 11.
3. Use the double angle identity for sine: \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \).
4. Calculate \( \sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{11}{61} \).
5. Calculate \( \cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{60}{61} \).

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, in a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is essential for solving problems involving angles and side lengths.

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For instance, the sine double angle formula states that sin(2α) = 2sin(α)cos(α). These formulas are useful for simplifying expressions and solving equations involving trigonometric functions, particularly when calculating values for angles that are multiples of a given angle.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that 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 finding missing side lengths in trigonometric problems and for verifying the relationships between the sides of a triangle.