Multiple Choice
Identify the most helpful first step in verifying the identity.
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.RE.10
Textbook Question
Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc θ - sin θ
Verified step by step guidance1
Recall that the cosecant function is the reciprocal of the sine function, so we can write \(\csc \theta\) as \(\frac{1}{\sin \theta}\).
Rewrite the expression \(\csc \theta - \sin \theta\) as \(\frac{1}{\sin \theta} - \sin \theta\) to express everything in terms of \(\sin \theta\).
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 \(\sin \theta\) and \(\cos \theta\) only, with no quotients involving other trigonometric functions.
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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 each other by expressing one as the reciprocal of another. For example, csc θ is the reciprocal of sin θ, meaning csc θ = 1/sin θ. Understanding these identities allows rewriting expressions to involve only sine and cosine.
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Pythagorean Identities
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves rewriting them to eliminate complex fractions or quotients and combining like terms. This process often uses algebraic manipulation and identities to express the function in a simpler or more standard form, such as removing quotients and using only sin θ and cos θ.
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Expressing Functions in Terms of sin θ and cos θ
Many trigonometric functions can be expressed using only sine and cosine, which are the fundamental trigonometric functions. For example, csc θ = 1/sin θ, sec θ = 1/cos θ, and tan θ = sin θ/cos θ. Writing expressions solely in terms of sin θ and cos θ helps in standardizing and simplifying trigonometric problems.
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