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 19

Chapter 3, Problem 19

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. sin 75° + sin 15°

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Recognize that the expression involves the sum of two sine functions: \(\sin 75^\circ + \sin 15^\circ\).

Recall the sum-to-product identity for sine functions: \(\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).

Apply the identity by setting \(A = 75^\circ\) and \(B = 15^\circ\), then calculate the averages: \(\frac{A+B}{2} = \frac{75^\circ + 15^\circ}{2}\) and \(\frac{A-B}{2} = \frac{75^\circ - 15^\circ}{2}\).

Rewrite the original sum as a product using the identity: \(2 \sin \left( \frac{75^\circ + 15^\circ}{2} \right) \cos \left( \frac{75^\circ - 15^\circ}{2} \right)\).

If required, evaluate the sine and cosine values at these angles to find the exact value of the product.

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Key Concepts

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

Sum-to-Product Formulas

Sum-to-product formulas convert sums or differences of sine or cosine functions into products, simplifying expressions. For sine, the formula sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) is used to rewrite sums as products, aiding in evaluation and further manipulation.

Exact Values of Special Angles

Certain angles like 15°, 30°, 45°, 60°, and 75° have known exact sine and cosine values derived from geometric constructions or half-angle formulas. Knowing these values allows precise calculation of trigonometric expressions without approximations.

Angle Addition and Subtraction

Understanding how to add and subtract angles is essential when applying sum-to-product formulas, as these require computing (A+B)/2 and (A−B)/2. This skill ensures correct substitution and simplification of trigonometric expressions.

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Textbook Question

Find all solutions of each equation. 2 cos x + √ 3 = 0

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Textbook Question

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 3x/2 + cos x/2

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Textbook Question

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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Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2cos² 𝝅/8﹣ 1

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