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 14

Chapter 3, Problem 14

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. sin(60° - 45°)

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Identify the sum or difference identity that applies to the expression. Since the expression is \( \sin(60^\circ - 45^\circ) \), use the sine difference identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).

Substitute \( A = 60^\circ \) and \( B = 45^\circ \) into the identity: \( \sin(60^\circ - 45^\circ) = \sin 60^\circ \cos 45^\circ - \cos 60^\circ \sin 45^\circ \).

Recall the exact values of the sine and cosine for the special angles: \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), \( \cos 60^\circ = \frac{1}{2} \), \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), and \( \cos 45^\circ = \frac{\sqrt{2}}{2} \).

Replace the trigonometric functions in the expression with their exact values: \( \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right) - \left( \frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right) \).

Simplify the expression by multiplying the fractions and combining like terms to find the exact value.

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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

Sum and difference 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 sine, the identity is sin(a - b) = sin(a)cos(b) - cos(a)sin(b), which allows exact evaluation of expressions like sin(60° - 45°).

Exact Values of Common Angles

Certain angles such as 30°, 45°, 60°, and their multiples have known exact sine and cosine values involving square roots. For example, sin(45°) = √2/2 and cos(60°) = 1/2. These values are essential for calculating exact trigonometric expressions without a calculator.

Angle Measurement in Degrees

Understanding that the angles given are in degrees is crucial for applying the correct trigonometric values and identities. Degrees are a common unit for measuring angles, and knowing how to work with degree measures ensures proper use of trigonometric formulas and exact values.

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Use the given information to find the exact value of each of the following: sin 2θ

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Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 4x + cos 2x

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Use a sum or difference formula to find the exact value of each expression. cos(45° + 30°)