Use one or more of the six sum and difference identities to solve Exercises 13–54.In Exercises 13–24, find the exact value of each expression.cos(135° + 30°)

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1. Identify the sum identity for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
2. Substitute \( a = 135^\circ \) and \( b = 30^\circ \) into the identity: \( \cos(135^\circ + 30^\circ) = \cos 135^\circ \cos 30^\circ - \sin 135^\circ \sin 30^\circ \).
3. Find \( \cos 135^\circ \) and \( \sin 135^\circ \) using the unit circle: \( \cos 135^\circ = -\frac{\sqrt{2}}{2} \) and \( \sin 135^\circ = \frac{\sqrt{2}}{2} \).
4. Find \( \cos 30^\circ \) and \( \sin 30^\circ \) using the unit circle: \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) and \( \sin 30^\circ = \frac{1}{2} \).
5. Substitute these values into the expression: \( \cos(135^\circ + 30^\circ) = \left(-\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) \).

Key Concepts

Sum and Difference Identities

The sum and difference identities are formulas that express the cosine and sine of the sum or difference of two angles in terms of the sine and cosine of those angles. For example, the cosine of the sum of two angles is given by cos(A + B) = cos(A)cos(B) - sin(A)sin(B). These identities are essential for simplifying expressions involving trigonometric functions of combined angles.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help in determining the values of trigonometric functions for angles greater than 90° or less than 0°. For instance, the reference angle for 135° is 45°, which is crucial for evaluating trigonometric functions in different quadrants.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for commonly used angles, such as 0°, 30°, 45°, 60°, and 90°. These values can be derived from the unit circle or special triangles. Knowing these exact values is vital for solving trigonometric expressions without a calculator.