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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 5 - Trigonometric IdentitiesProblema 73
Capitolo 6, Problema 73

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = (cot² x - 1)/(cot² x + 1)

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Start by recalling the double-angle identity for cosine: \(\cos 2x = \cos(x + x) = \cos^2 x - \sin^2 x\).
Express \(\cot x\) in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\), so \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\).
Rewrite the right-hand side of the equation \(\frac{\cot^2 x - 1}{\cot^2 x + 1}\) by substituting \(\cot^2 x\) with \(\frac{\cos^2 x}{\sin^2 x}\), giving \(\frac{\frac{\cos^2 x}{\sin^2 x} - 1}{\frac{\cos^2 x}{\sin^2 x} + 1}\).
Simplify the complex fraction by multiplying numerator and denominator by \(\sin^2 x\) to eliminate the denominators inside the fraction, resulting in \(\frac{\cos^2 x - \sin^2 x}{\cos^2 x + \sin^2 x}\).
Use the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\) to simplify the denominator, so the expression becomes \(\cos^2 x - \sin^2 x\), which matches the double-angle identity for \(\cos 2x\), verifying the identity.

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Double-Angle Identity for Cosine

The double-angle identity expresses cos 2x in terms of functions of x, such as cos 2x = cos² x - sin² x or cos 2x = 2 cos² x - 1. This identity helps rewrite trigonometric expressions involving 2x into simpler forms involving x, facilitating verification of equations.
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Double Angle Identities

Cotangent and Its Relationship to Sine and Cosine

Cotangent is defined as cot x = cos x / sin x. Understanding this relationship allows conversion of expressions involving cot² x into sine and cosine terms, which is essential for manipulating and simplifying the given equation.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Trigonometric Identities and Algebraic Manipulation

Verifying identities requires applying known trigonometric identities and algebraic techniques such as factoring, common denominators, and substitution. This process transforms one side of the equation to match the other, confirming the identity's validity.
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Fundamental Trigonometric Identities
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Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 72°

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sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

tan 72°

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Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

x + y = 9, 2x + y = -1

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Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = 1 - 2 sin² x

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Verify that each equation is an identity.

(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

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Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

sin 162°

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