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 67

Capitolo 6, Problema 67

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

Guida verificata passo dopo passo

Recall the angle addition formula for cosine: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\).

Apply the formula to the left side of the equation with \(a = \frac{\pi}{2}\) and \(b = x\): \(\cos\left(\frac{\pi}{2} + x\right) = \cos\frac{\pi}{2} \cos x - \sin\frac{\pi}{2} \sin x\).

Substitute the known values of \(\cos\frac{\pi}{2} = 0\) and \(\sin\frac{\pi}{2} = 1\) into the expression: \(0 \cdot \cos x - 1 \cdot \sin x\).

Simplify the expression to get \(-\sin x\) on the left side.

Since the right side of the original equation is \(-\sin x\), both sides are equal, 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. They allow the transformation and simplification of expressions, such as rewriting cos(π/2 + x) in terms of sine or cosine functions to verify equivalences.

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Percorso guidato

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Fundamental Trigonometric Identities

Angle Addition Formulas

Angle addition formulas express trigonometric functions of sums of angles, like cos(a + b) = cos a cos b - sin a sin b. These formulas are essential for breaking down complex expressions, such as cos(π/2 + x), into simpler components to facilitate verification of identities.

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Quadratic Formula

Relationship Between Sine and Cosine

Sine and cosine functions are closely related, often differing by phase shifts. For example, cos(π/2 + x) equals -sin x, showing how a cosine function shifted by π/2 relates directly to sine. Understanding this relationship helps in recognizing and proving trigonometric identities.

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Amplitude and Reflection of Sine and Cosine

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

Verify that each equation is an identity.

2 cos² (x/2) tan x = tan x+ sin x

719