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Chapter 5: Problem 12

Solve the equation. $$2 \sin x+1=0$$

Short Answer

Expert verified
The solution to the equation is \( x = \frac{7\pi}{6} , \frac{11\pi}{6} \).

Step by step solution

01

Isolate the Trigonometric Function

Start by isolating the trigonometric function, in this case the \( \sin x \). This can be done by subtracting 1 from both sides of the equation, which gives \( 2 \sin x = -1 \). Then, divide both sides by 2 to isolate \( \sin x \), which will give \( \sin x = -\frac{1}{2} \).
02

Use the Inverse Trigonometric Function

Next, we should recall that the inverse sine, noted as \( \sin^{-1} \) or arcsin, of a number is the angle whose sine is that number. In other words, \( x = \sin^{-1}\left(-\frac{1}{2}\right) \). However, since we're dealing with the unit circle, we should note that the sine function is negative in the third and fourth quadrants.
03

Identify The Angles

Since sine is negative in the third and fourth quadrants, and sine of \( \frac{\pi}{6} \) and \( \frac{5\pi}{6} \) is equal to \( \frac{1}{2} \), then \( x = \pi + \frac{\pi}{6} \) or \( x = 2\pi - \frac{\pi}{6} \), which gives \( x = \frac{7\pi}{6} \) and \( x = \frac{11\pi}{6} \).

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