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Chapter 5: Problem 40

Solve the multiple-angle equation. $$\sin 2 x=-\frac{\sqrt{3}}{2}$$

Short Answer

Expert verified
The principle solutions for the equation \(\sin 2x = -\frac{\sqrt{3}}{2}\) are \(x = 2\pi/3\) and \(x = 5\pi / 6\). The general solution to cover all possible answers is \(x = n\pi / 2 + (-1)^n\pi/6\).

Step by step solution

01

Identify the Quadrants for Solution

The given equation is \(\sin 2x = -\frac{\sqrt{3}}{2}\). The negative sign indicates that the solution lies either in the third or the fourth quadrant where the sine values are negative.
02

Determine the Reference Angle

The reference angle, \(\alpha\), is the angle for which \(\sin \alpha = \frac{\sqrt{3}}{2}\). This angle, from the unit circle, is \(60^{\circ}\) or \(\pi/3\) in radians.
03

Get the Principle Solution

For sine, the principle solution in third quadrant is \(\pi + \alpha\) and in the fourth quadrant is \(2\pi - \alpha\). Replacing the value of \(\alpha\), principle solutions are \(2x_1 = \pi + \pi/3 = 4\pi/3\) and \(2x_2 = 2\pi - \pi/3 = 5\pi/3\). Hence, \(x_1 = 2\pi/3\) and \(x_2 = 5\pi / 6\). These are the principle solutions for the equation.
04

Find all Solutions

The general solutions of a sine equation follow the pattern \(x =n\pi + (-1)^n\alpha\), where \(n\) is an integer. Applying this pattern, all solutions to this equation are \(x = n\pi / 2 + (-1)^n\pi/6\). This gives us not just the principle solutions but all possible solutions for the integer values of \(n\).

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Most popular questions from this chapter

Solve the multiple-angle equation. $$\sec 4 x=2$$

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