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 6.2.51

Capitolo 7, Problema 6.2.51

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
6 sin² θ + sin θ = 1

Guida verificata passo dopo passo

Rewrite the given equation in terms of \( \sin \theta \): \[ 6 \sin^{2} \theta + \sin \theta = 1 \].

Bring all terms to one side to form a quadratic equation in \( \sin \theta \): \[ 6 \sin^{2} \theta + \sin \theta - 1 = 0 \].

Let \( u = \sin \theta \). Substitute to get \( 6u^{2} + u - 1 = 0 \).

Solve the quadratic equation \( 6u^{2} + u - 1 = 0 \) using the quadratic formula: \[ u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a=6 \), \( b=1 \), and \( c=-1 \).

For each solution \( u \), check if it lies within the valid range for sine (\( -1 \leq u \leq 1 \)). Then solve for \( \theta \) by finding \( \theta = \arcsin(u) \) and use the unit circle to find all solutions within the specified domain, expressing answers in degrees and radians as required.

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Solving Quadratic Trigonometric Equations

This involves rewriting the trigonometric equation in a quadratic form, such as a sin²θ + b sinθ + c = 0, and then solving for the trigonometric function (e.g., sinθ) using algebraic methods like factoring or the quadratic formula.

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Percorso guidato

6:24

Solving Quadratic Equations by Completing the Square

General Solutions for Sine Equations

Once sinθ is found, the general solutions for θ are determined using the unit circle properties: θ = arcsin(value) + 2πn and θ = π - arcsin(value) + 2πn for integer n, ensuring all possible angles are considered.

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Percorso guidato

4:25

Introduction to Trig Equations

Converting Between Radians and Degrees and Using Least Nonnegative Angles

Understanding how to convert between radians and degrees is essential for expressing answers correctly. Additionally, solutions should be expressed as the smallest nonnegative angle within one full rotation (0 to 2π radians or 0° to 360°), rounding as specified.

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Percorso guidato

5:04

Converting between Degrees & Radians

Pratica correlata

Domanda del libro di testo

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

2√3 sin x/2 = 3

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Domanda del libro di testo

Solve for exact solutions over the interval [0°, 360°).

sin θ/2 = 0

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Domanda del libro di testo

Answer each question.

Suppose solving a trigonometric equation for solutions over the interval [0, 2π) leads to 2x = 2π/3, 2π, 8π/3. What are the corresponding values of x?

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Domanda del libro di testo

sin x (3 sin x - 1) = 1

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Domanda del libro di testo

Solve for exact solutions over the interval [0°, 360°).

sin θ/2 = -√3/2

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Domanda del libro di testo

cos θ/2 = 1

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