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 5.41
Simplify each expression.
±√[(1 - cos 8θ)/(1 + cos 8θ)]
Verified step by step guidance1
Recognize that the expression involves a square root of a fraction with trigonometric functions: \(\pm \sqrt{\frac{1 - \cos 8\theta}{1 + \cos 8\theta}}\).
Recall the trigonometric identity for tangent in terms of cosine: \(\tan^2 x = \frac{1 - \cos 2x}{1 + \cos 2x}\). Notice the similarity between this identity and the given expression.
Match the given expression to the identity by setting \(2x = 8\theta\), which implies \(x = 4\theta\). This allows rewriting the expression as \(\pm \sqrt{\tan^2 4\theta}\).
Since the square root of \(\tan^2 4\theta\) is the absolute value of \(\tan 4\theta\), and the expression already includes \(\pm\), simplify to \(\pm \tan 4\theta\).
Conclude that the simplified form of the original expression is \(\pm \tan 4\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Key identities like the Pythagorean identity and double-angle formulas help simplify complex expressions by rewriting them in more manageable forms.
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Fundamental Trigonometric Identities
Half-Angle Formulas
Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, sin²(θ) = (1 - cos 2θ)/2 and cos²(θ) = (1 + cos 2θ)/2. These formulas are essential for simplifying expressions involving ratios like (1 - cos α)/(1 + cos α).
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6:36Quadratic Formula
Simplification of Radical Expressions
Simplifying radical expressions involves rewriting square roots in simpler or more recognizable trigonometric forms. Recognizing patterns under the root and applying identities can transform complex radicals into basic trigonometric functions, often removing the radical sign.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
Factor each trigonometric expression.
sin³ α + cos³ α
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 sec θ
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