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Chapter 5: Problem 13

Solve the equation. $$\cos x+1=-\cos x$$

Short Answer

Expert verified
The solutions are \(x = 2\pi/3 + 2\pi k\) and \(x = -2\pi/3 + 2\pi k\) for any integer \(k\).

Step by step solution

01

Rearrange the equation

We need to start by moving all of the terms involving the cosine to one side of the equation and the constants to the other side. The equation becomes \(2\cos x = -1\)
02

Solve the equation for \(x\)

Now, we divide by 2 to isolate \(\cos x\) on one side. The equation becomes \(\cos x = -1/2\). We know that cosine function has the value of -1/2 at \(2\pi/3\) radians and \(-2\pi/3\) radians in a standard unit circle.
03

Generalize the solution for all values

Because the cosine function has a period of \(2\pi\), we can generalize all solutions with the following form: \(x = 2\pi/3 + 2\pi k\) and \(x = -2\pi/3 + 2\pi k\) where \(k\) is any integer.

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