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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 5.RE.10
Chapter 6, Problem 5.RE.10

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 guidance
1
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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Related Practice
Textbook Question

Find values of the sine and cosine functions for each angle measure.

2y, given sec y = -5/3, sin y > 0

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Textbook Question

Use the given information to cos(x - y).

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Textbook Question

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(1 - cos 2x)/sin 2x

Textbook Question

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos 75°

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For each expression in Column I, choose the expression from Column II that completes an identity.

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D. csc ^2 x-cot ^2 x + sin ^2 x

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Textbook Question

Work each problem.

Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.