Textbook Question
In Exercises 39–46, use a half-angle formula to find the exact value of each expression. sin 105°
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.3.37
In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. sin² x cos² x
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Recall the power-reducing formulas for sine squared and cosine squared:
\(\sin^{2}x = \frac{1 - \cos(2x)}{2}\)
and
\(\cos^{2}x = \frac{1 + \cos(2x)}{2}\).
Rewrite the expression \(\sin^{2}x \cos^{2}x\) by substituting the power-reducing formulas:
\(\sin^{2}x \cos^{2}x = \left(\frac{1 - \cos(2x)}{2}\right) \times \left(\frac{1 + \cos(2x)}{2}\right)\).
Multiply the two fractions:
\(\sin^{2}x \cos^{2}x = \frac{(1 - \cos(2x))(1 + \cos(2x))}{4}\).
Recognize that \((1 - \cos(2x))(1 + \cos(2x))\) is a difference of squares:
\(1 - \cos^{2}(2x)\).
Use the Pythagorean identity \(\sin^{2}\theta = 1 - \cos^{2}\theta\) to rewrite the expression as:
\(\sin^{2}x \cos^{2}x = \frac{\sin^{2}(2x)}{4}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power-Reducing Formulas
Power-reducing formulas express powers of sine and cosine functions in terms of first powers of trigonometric functions with multiple angles. For example, sin²x can be rewritten as (1 - cos 2x)/2. These formulas simplify expressions by reducing the exponent, making integration and other operations easier.
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Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)
Double-Angle Identities
Double-angle identities relate trigonometric functions of double angles to functions of single angles, such as cos 2x = cos²x - sin²x. These identities are essential for applying power-reducing formulas because they allow rewriting powers of sine and cosine in terms of cos 2x or sin 2x.
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05:06Double Angle Identities
Product-to-Sum Formulas
Product-to-sum formulas convert products of sine and cosine functions into sums or differences of trigonometric functions. For example, sin A cos B = (1/2)[sin(A+B) + sin(A-B)]. These formulas help simplify expressions like sin²x cos²x by breaking down products into sums, facilitating further reduction.
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2:25Verifying Identities with Sum and Difference Formulas
Related Practice
Textbook Question
Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 9 tan² x - 3 = 0
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Textbook Question
Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan 3x = (√3)/3
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Textbook Question
In Exercises 39–46, use a half-angle formula to find the exact value of each expression. tan(7𝝅/8)
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Textbook Question
Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sin² x = sin x + 3
Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. sin(θ/2)
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