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Chapter 5: Problem 43

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

Short Answer

Expert verified
The solutions are \(x = \frac{\pi}{2} + 2n\pi\) and \(x = \frac{7\pi}{2} + 2n\pi\), where \(n\) is any integer.

Step by step solution

01

Identify correspondance with unit circle value

The value \(\frac{\sqrt{2}}{2}\) corresponds to \(\frac{\pi}{4}\) and \(\frac{7\pi}{4}\) in the unit circle of cosine function. Therefore, the angle \(\frac{x}{2}\) can be \(\frac{\pi}{4}\) or \(\frac{7\pi}{4}\).
02

Solve for x

Since \(\frac{x}{2}\) can be \(\frac{\pi}{4}\) or \(\frac{7\pi}{4}\), Solve for x to get \(x = 2\left( \frac{\pi}{4}\right) = \frac{\pi}{2}\) and \(x = 2\left( \frac{7\pi}{4}\right) = \frac{7\pi}{2}\).
03

General Solution

Considering that cosine function has a period of \(2\pi\), a general solution which is the set of all possible solutions, for x is \(\frac{\pi}{2} + 2n\pi\) and \(\frac{7\pi}{2} + 2n\pi\), where \(n\) is any integer. This is because adding any integral multiple of \(2\pi\) to the angle does not change the value of the cosine function.

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