In Exercises 63–68, find the exact value of each expression. Do not use a calculator.cos 12° sin 78° + cos 78° sin 12°

Verified step by step guidance

1. Recognize the expression cos 12° sin 78° + cos 78° sin 12° as a trigonometric identity.
2. Recall the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
3. Identify that the given expression matches the sine addition formula with \( A = 12° \) and \( B = 78° \).
4. Substitute \( A = 12° \) and \( B = 78° \) into the sine addition formula: \( \sin(12° + 78°) \).
5. Simplify the expression to find \( \sin(90°) \), knowing that \( \sin(90°) \) has a well-known exact value.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. A key identity relevant to this question is the sine addition formula, which states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). This identity allows us to simplify expressions involving sums of angles.

Sine and Cosine Functions

The sine and cosine functions are fundamental trigonometric functions that relate the angles of a triangle to the ratios of its sides. For any angle θ, sin(θ) represents the ratio of the opposite side to the hypotenuse, while cos(θ) represents the ratio of the adjacent side to the hypotenuse. Understanding these functions is crucial for evaluating trigonometric expressions.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often expressed in terms of square roots or fractions. For angles like 12° and 78°, knowing their exact values or how to derive them using known angles (like 30°, 45°, and 60°) is essential for solving trigonometric expressions without a calculator.