Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.RE.48

Capitolo 6, Problema 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

Guida verificata passo dopo passo

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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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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Quotients of Complex Numbers in Polar Form

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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Fundamental Trigonometric Identities

Pratica correlata

Domanda del libro di testo

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 - cos θ) (csc θ + sec θ)

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Domanda del libro di testo

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.

csc² θ + sec² θ

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Domanda del libro di testo

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

6. sec² x = ____

A. sin ^2 x/cos ^2 x