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.
2 cos² (x/2) tan x = tan x+ sin x

Guida verificata passo dopo passo

Start by writing down the given equation to verify: \(2 \cos^{2}\left(\frac{x}{2}\right) \tan x = \tan x + \sin x\).

Recall the double-angle identity for cosine: \(\cos x = 2 \cos^{2}\left(\frac{x}{2}\right) - 1\). From this, express \(2 \cos^{2}\left(\frac{x}{2}\right)\) as \(1 + \cos x\).

Substitute \(2 \cos^{2}\left(\frac{x}{2}\right)\) with \(1 + \cos x\) in the left side of the equation, so it becomes \((1 + \cos x) \tan x\).

Rewrite \(\tan x\) as \(\frac{\sin x}{\cos x}\) and distribute on the left side: \((1 + \cos x) \cdot \frac{\sin x}{\cos x} = \frac{\sin x}{\cos x} + \frac{\sin x \cos x}{\cos x}\).

Simplify the right side of the expression by canceling \(\cos x\) in the second term, resulting in \(\frac{\sin x}{\cos x} + \sin x\), which matches the right side of the original equation, confirming 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 Pythagorean or angle-sum identities.

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Fundamental Trigonometric Identities

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, cos²(x/2) can be rewritten using the identity cos²(θ) = (1 + cos(2θ))/2, which helps simplify expressions involving cos²(x/2).

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Quadratic Formula

Relationship Between Tangent, Sine, and Cosine

Tangent is defined as sin(x)/cos(x). Understanding this relationship allows rewriting tan x in terms of sine and cosine, facilitating simplification and comparison of both sides of the equation when verifying identities.

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

Pratica correlata

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Verify that each equation is an identity.

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(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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