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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.48
Chapter 6, Problem 5.RE.48

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

Verified step by step guidance
1
Recall the definitions of the cosecant and cotangent functions in terms of sine and cosine: \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\).
Rewrite the expression \(\csc x - \cot x\) using these definitions: \(\frac{1}{\sin x} - \frac{\cos x}{\sin x}\).
Combine the terms over the common denominator \(\sin x\): \(\frac{1 - \cos x}{\sin x}\).
Graph the original expression \(\csc x - \cot x\) and the simplified expression \(\frac{1 - \cos x}{\sin x}\) over a suitable domain (avoiding points where \(\sin x = 0\)) to observe if they coincide, which suggests an identity.
To verify algebraically, try to manipulate \(\frac{1 - \cos x}{\sin x}\) into a known trigonometric expression or use conjugates to simplify and confirm the identity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Reciprocal and Quotient Identities

Understanding that cosecant (csc x) is the reciprocal of sine (1/sin x) and cotangent (cot x) is the quotient of cosine over sine (cos x/sin x) is essential. These identities allow rewriting expressions involving csc and cot in terms of sine and cosine, facilitating algebraic manipulation and verification.
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Graphing Trigonometric Functions

Graphing functions like csc x and cot x helps visualize their behavior, including asymptotes and periodicity. By plotting csc x - cot x, one can observe patterns or simplifications that suggest possible identities, providing an intuitive basis before algebraic proof.
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Introduction to Trigonometric Functions

Algebraic Verification of Trigonometric Identities

After conjecturing an identity from the graph, algebraic verification involves manipulating one side of the equation using known identities to prove equality. This process requires skill in factoring, common denominators, and applying fundamental trigonometric identities to confirm the conjecture rigorously.
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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

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

4. cot x = ____


II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

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

E. tan 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.

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Textbook Question

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

6. sec² x = ____


II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

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

E. tan x

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Textbook Question

Use the given information to find sin(x + y).

sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III

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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.

(cos x sin 2x)/1 + cos 2x)

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