Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 17

Chapter 3, Problem 17

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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