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 6.3.23

Capitolo 7, Problema 6.3.23

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

√2 cos 2θ = -1

Guida verificata passo dopo passo

Start with the given equation: \(\sqrt{2} \cos 2\theta = -1\).

Isolate the cosine term by dividing both sides by \(\sqrt{2}\): \(\cos 2\theta = \frac{-1}{\sqrt{2}}\).

Recognize that \(\cos 2\theta = -\frac{1}{\sqrt{2}}\) corresponds to specific angles where the cosine value is negative and equal to \(-\frac{1}{\sqrt{2}}\).

Find the general solutions for \(2\theta\) by identifying the reference angle whose cosine is \(\frac{1}{\sqrt{2}}\), which is \(45^\circ\) or \(\frac{\pi}{4}\) radians, and then determine the angles in the appropriate quadrants where cosine is negative (quadrants II and III).

Divide the solutions for \(2\theta\) by 2 to solve for \(\theta\), and then restrict the solutions to the given interval \([0^\circ, 360^\circ)\) or \([0, 2\pi)\) depending on the unit.

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Double-Angle Trigonometric Functions

The equation involves cos 2θ, a double-angle function representing the cosine of twice the angle θ. Understanding how to manipulate and solve equations with double angles is essential, as it changes the period and affects the solution set within the given interval.

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Double Angle Identities

Solving Trigonometric Equations

Solving trigonometric equations requires isolating the trigonometric function and finding all angles within the specified interval that satisfy the equation. This often involves using inverse trigonometric functions and considering the periodic nature of sine and cosine.

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How to Solve Linear Trigonometric Equations

Interval and Exact Solutions in Radians and Degrees

The problem specifies solutions over [0, 2π) for x and [0°, 360°) for θ, requiring careful attention to the domain. Providing exact solutions means expressing answers in terms of π or degrees rather than decimal approximations, ensuring precision and completeness.

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Converting between Degrees & Radians

Pratica correlata

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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 cos² x + cos x ― 1 = 0

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√2 sin 3x - 1 = 0

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cos θ + 1 = 0

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

tan 2x + sec 2x = 3

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