Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
csc θ cos θ tan θ
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.32
Factor each trigonometric expression.
sin³ α + cos³ α
Verified step by step guidance1
Recognize that the expression \( \sin^3 \alpha + \cos^3 \alpha \) is a sum of cubes.
Recall the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).
Identify \( a = \sin \alpha \) and \( b = \cos \alpha \) in the expression.
Apply the sum of cubes formula: \( \sin^3 \alpha + \cos^3 \alpha = (\sin \alpha + \cos \alpha)(\sin^2 \alpha - \sin \alpha \cos \alpha + \cos^2 \alpha) \).
Simplify the second factor using the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \), resulting in \( 1 - \sin \alpha \cos \alpha \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sum of Cubes Formula
The sum of cubes formula states that a³ + b³ can be factored as (a + b)(a² - ab + b²). This formula is essential for factoring expressions like sin³ α + cos³ α, where a = sin α and b = cos α. Understanding this formula allows us to break down complex trigonometric expressions into simpler components.
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Verifying Identities with Sum and Difference Formulas
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. Key identities, such as sin² α + cos² α = 1, can be useful when simplifying or manipulating trigonometric expressions. Recognizing these identities helps in transforming and factoring expressions effectively.
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5:32Fundamental Trigonometric Identities
Factoring Techniques
Factoring techniques involve rewriting an expression as a product of its factors. This includes recognizing patterns, such as the sum of cubes, and applying algebraic methods to simplify expressions. Mastery of these techniques is crucial for solving trigonometric problems and simplifying complex expressions.
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6:08Factoring
Related Practice
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sin θ - cos θ) (csc θ + sec θ)
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Textbook Question
Simplify each expression.
±√[(1 - cos 8θ)/(1 + cos 8θ)]
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
[1 - sin²(-θ)]/[1 + cot²(-θ)]
Textbook Question
Perform each indicated operation and simplify the result so that there are no quotients.
(1 + tan θ)² - 2 tan θ
Textbook Question
Verify that each equation is an identity.
sec⁴ x - sec² x = tan⁴ x + tan² x