Textbook Question
Verify that each equation is an identity.
sin³ θ = sin θ - cos² θ sin θ
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.1.68
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 - sin θ
Verified step by step guidance1
Recall the definition of cosecant in terms of sine: \(\csc \theta = \frac{1}{\sin \theta}\).
Rewrite the expression \(\csc \theta - \sin \theta\) by substituting \(\csc \theta\) with \(\frac{1}{\sin \theta}\), so it becomes \(\frac{1}{\sin \theta} - \sin \theta\).
To combine the terms into a single expression without quotients, find a common denominator, which is \(\sin \theta\), and rewrite the expression as \(\frac{1}{\sin \theta} - \frac{\sin^2 \theta}{\sin \theta}\).
Combine the fractions over the common denominator: \(\frac{1 - \sin^2 \theta}{\sin \theta}\).
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to replace \(1 - \sin^2 \theta\) with \(\cos^2 \theta\), resulting in \(\frac{\cos^2 \theta}{\sin \theta}\). This expression is now in terms of sine and cosine, but still has a quotient; to eliminate the quotient, express it as \(\cos \theta \cdot \frac{\cos \theta}{\sin \theta}\) or \(\cos \theta \cdot \cot \theta\), depending on the problem's requirements.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Identities
Reciprocal identities relate trigonometric functions to their reciprocals, such as cosecant (csc θ) being the reciprocal of sine (sin θ). Specifically, csc θ = 1/sin θ. Understanding these identities allows rewriting expressions involving csc θ in terms of sin θ.
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Pythagorean Identities
Expression Simplification
Simplification involves rewriting expressions to eliminate complex fractions or quotients, often by finding a common denominator or multiplying through by terms. The goal is to express the function using only sine and cosine without any division, making the expression easier to interpret or use.
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Trigonometric Functions in Terms of θ
Trigonometric expressions should be written solely in terms of the angle θ, avoiding other variables or functions. This ensures clarity and consistency, especially when simplifying or combining functions, and helps in applying further identities or solving equations.
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