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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Identify the sum identity for cosine, which is: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).

Assign the given angles to the variables: let \(A = 135^\circ\) and \(B = 30^\circ\).

Substitute the values into the sum identity formula: \(\cos(135^\circ + 30^\circ) = \cos 135^\circ \cos 30^\circ - \sin 135^\circ \sin 30^\circ\).

Recall or find the exact values of the trigonometric functions for these special angles: \(\cos 135^\circ\), \(\cos 30^\circ\), \(\sin 135^\circ\), and \(\sin 30^\circ\).

Plug these exact values into the expression and simplify step-by-step to find the exact value of \(\cos(165^\circ)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sum and Difference Identities

These identities express the sine, cosine, or tangent of a sum or difference of two angles in terms of the sines and cosines of the individual angles. For cosine, the sum identity is cos(A + B) = cos A cos B - sin A sin B, which allows exact evaluation of angles not commonly found on the unit circle.

Exact Values of Trigonometric Functions for Special Angles

Certain angles like 30°, 45°, 60°, and their multiples have known exact sine and cosine values involving square roots. Knowing these values is essential to compute expressions like cos(135° + 30°) exactly without a calculator.

Angle Addition in Degrees

Understanding how to add angles measured in degrees and apply trigonometric identities correctly is crucial. Here, 135° + 30° equals 165°, and using the sum identity requires substituting the correct angle measures and their trigonometric values.

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