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

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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Recognize that the expression involves the sum of two cosine terms: \(\cos\left(\frac{3x}{2}\right) + \cos\left(\frac{x}{2}\right)\). This suggests using the cosine sum-to-product identity.

Recall the cosine sum-to-product formula: \(\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\).

Identify \(A = \frac{3x}{2}\) and \(B = \frac{x}{2}\), then compute the average and difference inside the cosines: \(\frac{A+B}{2} = \frac{\frac{3x}{2} + \frac{x}{2}}{2}\) and \(\frac{A-B}{2} = \frac{\frac{3x}{2} - \frac{x}{2}}{2}\).

Simplify these expressions to get the arguments for the product form: \(\cos\left(\frac{A+B}{2}\right)\) and \(\cos\left(\frac{A-B}{2}\right)\).

Write the original sum as the product \(2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\), which expresses the sum of cosines as a 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 trigonometric functions into products, simplifying expressions and solving equations. For example, cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2). These formulas are essential for rewriting sums or differences as products.

Evaluating Trigonometric Expressions

After rewriting expressions using sum-to-product identities, evaluating the exact value involves substituting known angle values and using unit circle values or special angle properties. This step ensures the expression is simplified to a precise numerical value.

Angle Manipulation and Simplification

Understanding how to manipulate angles, such as adding, subtracting, or halving angles, is crucial when applying sum-to-product formulas. Simplifying angles to standard positions or known reference angles helps in both rewriting and evaluating trigonometric expressions.

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Related Practice

Textbook Question

Use a sum or difference formula to find the exact value of each expression. tan 5𝝅/12

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. sin 75° + sin 15°

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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Find all solutions of each equation. tan x = 0

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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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