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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.3.21
Capitolo 7, Problema 6.3.21

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.


3 tan 3x = √3

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1
Start by isolating the tangent function in the equation: given \(3 \tan 3x = \sqrt{3}\), divide both sides by 3 to get \(\tan 3x = \frac{\sqrt{3}}{3}\).
Recall the exact values of tangent for common angles. Since \(\tan \theta = \frac{\sqrt{3}}{3}\), identify the reference angle \(\alpha\) such that \(\tan \alpha = \frac{\sqrt{3}}{3}\). This corresponds to \(\alpha = \frac{\pi}{6}\) radians or 30°.
Write the general solution for \(3x\) using the periodicity of the tangent function, which has period \(\pi\). The solutions are given by \(3x = \alpha + k\pi\), where \(k\) is any integer.
Express \(x\) explicitly by dividing both sides by 3: \(x = \frac{\alpha}{3} + \frac{k\pi}{3}\).
Find all values of \(x\) within the interval \([0, 2\pi)\) by substituting integer values of \(k\) such that \(x\) remains in the interval. List these exact solutions.

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Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values within a given interval that satisfy the equation. This often requires using inverse trigonometric functions and considering the periodic nature of trig functions to find multiple solutions.
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How to Solve Linear Trigonometric Equations

Properties of the Tangent Function

The tangent function, tan(θ), has a period of π, meaning its values repeat every π radians. It is undefined at odd multiples of π/2 and can take any real value. Understanding its periodicity is essential for finding all solutions within a specified interval.
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Introduction to Tangent Graph

Interval Notation and Angle Measurement

The problem specifies solutions over intervals [0, 2π) for radians and [0°, 360°) for degrees. Knowing how to convert between degrees and radians and interpreting these intervals correctly ensures that all valid solutions are found and expressed within the required domain.
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i & j Notation
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Solve each equation for all exact solutions, in degrees.

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Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

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Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.


sin 3θ = -1

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Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.


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sin θ cos θ ― sin θ = 0

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