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 that the expression \( \cos 12^\circ \sin 78^\circ + \cos 78^\circ \sin 12^\circ \) matches the form of the sine addition formula, which is \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
2. Identify the angles in the expression to match the formula: let \( A = 78^\circ \) and \( B = 12^\circ \), so the expression becomes \( \cos 12^\circ \sin 78^\circ + \cos 78^\circ \sin 12^\circ = \sin(78^\circ + 12^\circ) \).
3. Add the angles inside the sine function: \( 78^\circ + 12^\circ = 90^\circ \), so the expression simplifies to \( \sin 90^\circ \).
4. Recall the exact value of \( \sin 90^\circ \), which is a fundamental trigonometric value.
5. Conclude that the original expression is equal to \( \sin 90^\circ \), and thus find the exact value without using a calculator.

Key Concepts

Sum of Angles Identity for Sine

The sum of angles identity states that sin(A + B) = sin A cos B + cos A sin B. This formula allows the expression cos 12° sin 78° + cos 78° sin 12° to be recognized as sin(12° + 78°), simplifying the calculation without a calculator.

Exact Values of Trigonometric Functions

Exact values refer to the precise trigonometric values for special angles, often expressed in fractions or radicals. Knowing these values or how to simplify expressions to standard angles like 90° helps find exact results without decimal approximations.

Angle Complementarity in Trigonometry

Complementary angles sum to 90°, and their sine and cosine values are related: sin θ = cos(90° - θ). Recognizing complementary angles can simplify expressions and help verify results, especially when dealing with angles like 12° and 78°.