Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problem 6.3.47

Chapter 7, Problem 6.3.47

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 - sin 2θ = 4 sin 2θ

Verified step by step guidance

Start by rewriting the given equation: \(2 - \sin 2\theta = 4 \sin 2\theta\).

Combine like terms by adding \(\sin 2\theta\) to both sides to isolate the sine term: \(2 = 4 \sin 2\theta + \sin 2\theta\) which simplifies to \(2 = 5 \sin 2\theta\).

Solve for \(\sin 2\theta\) by dividing both sides by 5: \(\sin 2\theta = \frac{2}{5}\).

Find the general solutions for \(2\theta\) by using the inverse sine function: \(2\theta = \sin^{-1}\left(\frac{2}{5}\right)\) and also consider the second solution in the range \([0, 2\pi)\), which is \(2\theta = \pi - \sin^{-1}\left(\frac{2}{5}\right)\).

Divide all solutions for \(2\theta\) by 2 to solve for \(\theta\), then express the answers in radians and degrees. For approximate answers, round radians to four decimal places and degrees to the nearest tenth. Also, write the solutions using the least possible nonnegative angle measures.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Trigonometric Equations

This involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. Solutions often include general forms using periodicity, and exact or approximate values depending on the problem's requirements.

Double-Angle Identities

Double-angle identities express trigonometric functions of 2θ in terms of θ, such as sin(2θ) = 2 sin θ cos θ. Recognizing and using these identities helps simplify equations and find solutions more efficiently.

Angle Measurement and Conversion

Understanding the difference between radians and degrees, and converting between them, is essential. Solutions must be expressed in the correct units, rounded appropriately, and within the specified interval, often using the least nonnegative angle measure.

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