In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°

Verified step by step guidance

1. Recognize the double angle identity for sine: $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$.
2. Identify that the given expression $2 \sin(15^\circ) \cos(15^\circ)$ matches the form of the double angle identity for sine.
3. Rewrite the expression $2 \sin(15^\circ) \cos(15^\circ)$ as $\sin(2 \times 15^\circ)$.
4. Simplify the expression to $\sin(30^\circ)$ using the double angle identity.
5. Use the known value of $\sin(30^\circ)$ to find the exact value of the expression.

Key Concepts

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For sine, cosine, and tangent, these formulas are: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) - sin²(θ), and tan(2θ) = 2tan(θ)/(1 - tan²(θ)). Understanding these formulas is essential for simplifying expressions involving double angles.

Sine and Cosine Values

The sine and cosine functions are fundamental in trigonometry, representing the ratios of the sides of a right triangle. For specific angles, such as 15°, these values can be derived using known identities or approximations. Recognizing these values is crucial for evaluating expressions and applying the double angle formulas effectively.

Exact Values vs. Approximate Values

Exact values in trigonometry refer to precise numerical representations of trigonometric functions, often expressed in terms of radicals or fractions, rather than decimal approximations. For example, sin(15°) and cos(15°) can be expressed using the half-angle or sum formulas. Understanding how to derive and use these exact values is important for solving trigonometric problems accurately.