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 66

Capitolo 6, Problema 66

Verify that each equation is an identity.
sin(s + t)/cos s cot t = tan s + tan t

Guida verificata passo dopo passo

Start by writing down the given equation to verify: \(\frac{\sin(s + t)}{\cos s} \cot t = \tan s + \tan t\).

Recall the angle sum identity for sine: \(\sin(s + t) = \sin s \cos t + \cos s \sin t\). Substitute this into the left-hand side (LHS) of the equation.

Rewrite \(\cot t\) as \(\frac{\cos t}{\sin t}\) and substitute it into the LHS, so the expression becomes \(\frac{\sin s \cos t + \cos s \sin t}{\cos s} \times \frac{\cos t}{\sin t}\).

Simplify the LHS by distributing and canceling terms where possible. Break the fraction into two parts: \(\frac{\sin s \cos t}{\cos s} \times \frac{\cos t}{\sin t} + \frac{\cos s \sin t}{\cos s} \times \frac{\cos t}{\sin t}\).

Express the resulting terms in terms of tangent functions, using \(\tan x = \frac{\sin x}{\cos x}\), and simplify to show that the LHS equals the right-hand side (RHS), \(\tan s + \tan t\), thus 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, often by using fundamental identities like Pythagorean, reciprocal, or quotient identities.

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Percorso guidato

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Fundamental Trigonometric Identities

Quotient and Reciprocal Identities

Quotient identities express tangent and cotangent in terms of sine and cosine: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Reciprocal identities relate sine, cosine, and their reciprocals cosecant and secant. These are essential for rewriting expressions to simplify or prove identities.

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Quotients of Complex Numbers in Polar Form

Sum of Angles Formulas

Sum of angles formulas provide expressions for trigonometric functions of sums, such as sin(s + t) = sin s cos t + cos s sin t. These formulas are crucial for expanding or simplifying expressions involving sums of angles to verify identities.

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Verifying Identities with Sum and Difference Formulas

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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.

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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.

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

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