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 68

Capitolo 6, Problema 68

Verify that each equation is an identity.
sin(x + y)/cos(x - y) = (cot x + cot y)/(1 + cot x cot y)

Guida verificata passo dopo passo

Start by recalling the definitions of cotangent in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\) and \(\cot y = \frac{\cos y}{\sin y}\).

Rewrite the right-hand side (RHS) of the equation \(\frac{\cot x + \cot y}{1 + \cot x \cot y}\) by substituting the cotangent expressions: \(\frac{\frac{\cos x}{\sin x} + \frac{\cos y}{\sin y}}{1 + \frac{\cos x}{\sin x} \cdot \frac{\cos y}{\sin y}}\).

Find a common denominator for the numerator and denominator of the RHS to combine the fractions: numerator becomes \(\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y}\) and denominator becomes \(\frac{\sin x \sin y + \cos x \cos y}{\sin x \sin y}\).

Simplify the complex fraction by multiplying numerator and denominator by \(\sin x \sin y\), which will cancel the denominators inside the fraction, resulting in \(\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y + \cos x \cos y}\).

Recognize the trigonometric sum formulas: numerator \(\cos x \sin y + \cos y \sin x = \sin(x + y)\) and denominator \(\sin x \sin y + \cos x \cos y = \cos(x - y)\). Thus, the RHS simplifies to \(\frac{\sin(x + y)}{\cos(x - y)}\), which matches the left-hand side (LHS), verifying the identity.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression using known formulas, such as angle sum and difference identities.

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Percorso guidato

5:32

Fundamental Trigonometric Identities

Angle Sum and Difference Formulas

These formulas express trigonometric functions of sums or differences of angles in terms of functions of individual angles. For example, sin(x + y) = sin x cos y + cos x sin y and cos(x - y) = cos x cos y + sin x sin y, which are essential for rewriting and simplifying expressions.

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Percorso guidato

2:25

Verifying Identities with Sum and Difference Formulas

Cotangent and Its Relationship to Sine and Cosine

Cotangent is the reciprocal of tangent, defined as cot θ = cos θ / sin θ. Understanding how to rewrite cotangent in terms of sine and cosine helps in transforming and simplifying expressions involving cot x and cot y to verify identities.

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Percorso guidato

5:08

Sine, Cosine, & Tangent of 30°, 45°, & 60°

Pratica correlata

Domanda del libro di testo

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.

cot 18°

757

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Domanda del libro di testo

Verify that each equation is an identity.

(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

733

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Domanda del libro di testo

Write each expression as a product of trigonometric functions. See Example 8.

sin 9x - sin 3x

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