Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 9

Capitolo 7, Problema 9

Solve each equation for x, where x is restricted to the given interval.
y = 3 tan 2x , for x in [―π/4, π/4]

Guida verificata passo dopo passo

Rewrite the given equation clearly: $y = 3 \tan(2x)$. To solve for $x$, we need to isolate $x$ in terms of $y$.

Divide both sides of the equation by 3 to isolate the tangent function: $\tan(2x) = \frac{y}{3}$.

Apply the inverse tangent (arctangent) function to both sides to solve for \$2x$: $2x = \arctan\left(\frac{y}{3}\right)$.

Divide both sides by 2 to solve for $x$: $x = \frac{1}{2} \arctan\left(\frac{y}{3}\right)$.

Since $x$ is restricted to the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, check that the values of $x$ obtained from the inverse tangent fall within this interval, considering the periodicity and range of the tangent function.

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Properties of the Tangent Function

The tangent function, tan(θ), is periodic with period π and has vertical asymptotes where cos(θ) = 0. Understanding its behavior, including its range and points of discontinuity, is essential for solving equations involving tangent, especially when restricting the domain.

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Introduction to Tangent Graph

Solving Trigonometric Equations

Solving equations like y = 3 tan(2x) involves isolating the trigonometric function and using inverse functions to find general solutions. One must consider the periodicity of tangent and apply domain restrictions to identify all valid solutions within the given interval.

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How to Solve Linear Trigonometric Equations

Domain and Interval Restrictions

When solving trigonometric equations, restricting the variable x to a specific interval, such as [−π/4, π/4], limits the possible solutions. It is crucial to check which solutions fall within this interval to ensure the answer set is accurate and complete.

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Percorso guidato

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Finding the Domain of an Equation

Pratica correlata

Domanda del libro di testo

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

sin² θ + 3 sin θ + 2 = 0

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Domanda del libro di testo

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).

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cos θ = ―1/2

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Domanda del libro di testo

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

3 cos² θ + 2 cos θ - 1 = 0

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Domanda del libro di testo

Find the exact value of each real number y. Do not use a calculator.

y = cos⁻¹ (―√2/2)

697

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