Ch. 3 - Trigonometric Identities and Equations

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Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problema 3.RE.62

Capitolo 3, Problema 3.RE.62

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x = √ 3 sin x

Guida verificata passo dopo passo

Start by rewriting the given equation: \(\sin 2x = \sqrt{3} \sin x\).

Use the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\). Substitute this into the equation to get \(2 \sin x \cos x = \sqrt{3} \sin x\).

Bring all terms to one side: \(2 \sin x \cos x - \sqrt{3} \sin x = 0\). Factor out \(\sin x\): \(\sin x (2 \cos x - \sqrt{3}) = 0\).

Set each factor equal to zero and solve separately: 1) \(\sin x = 0\) 2) \(2 \cos x - \sqrt{3} = 0\).

For \(\sin x = 0\), find all \(x\) in \([0, 2\pi)\) where sine is zero. For \(2 \cos x - \sqrt{3} = 0\), solve for \(\cos x = \frac{\sqrt{3}}{2}\) and find all \(x\) in \([0, 2\pi)\) that satisfy this.

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Double-Angle Identity for Sine

The double-angle identity expresses sin(2x) as 2 sin(x) cos(x). This allows rewriting the equation sin 2x = √3 sin x into a form involving sin(x) and cos(x), facilitating algebraic manipulation and solution finding.

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Percorso guidato

05:06

Double Angle Identities

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the given interval. It often requires factoring, using identities, and considering the periodic nature of sine and cosine functions.

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Percorso guidato

4:34

How to Solve Linear Trigonometric Equations

Interval Restriction and Exact Values

Solutions must be found within the interval [0, 2π), meaning all valid angles between 0 and 2π are considered. Using exact values (like π/3, π/6) is preferred, but approximate decimal values to four decimal places are acceptable when exact forms are complex.

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Percorso guidato

5:10

Evaluate Composite Functions - Values on Unit Circle

Pratica correlata

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In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°

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In Exercises 35–38, find the exact value of the following under the given conditions:

b. cos(α﹣β)

sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

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In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos 2x = -1

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Domanda del libro di testo

In Exercises 35–38, find the exact value of the following under the given conditions:

a. sin(α + β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

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Domanda del libro di testo

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

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