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Chapter 5: Problem 42

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

Short Answer

Expert verified
The solutions to the given trigonometric equation are \(x = \pi/12 + k\pi/2\) and \(x = -\pi/12 + k\pi/2\) where k is an integer.

Step by step solution

01

Express secant in terms of cosine

The secant function is the reciprocal of the cosine function. Therefore, we can write the equation as \(\frac{1}{\cos 4x}=2\)
02

Solve for \cos$

Next, solve the equation for \cos. This is done by taking the reciprocal of both sides to get \(\cos 4x = \frac{1}{2}\)
03

Solve for the angle

Now, we need to find the value of the angle 4x that gives a cosine value of 1/2. From the unit circle, we know that cosine equals 1/2 at angles \(\pi/3 + 2k\pi\) and \(-\pi/3 +2k\pi\) where k is an integer. So, \(4x = \pi/3 + 2k\pi\) or \(4x = -\pi/3 + 2k\pi\). Hence the solutions for x are: \(x = \pi/12 + k\pi/2\) and \(x = -\pi/12 + k\pi/2\)
04

Check if solutions fall within the domain

Based on the problem, ensure that the solutions for x are within given domain. If no domain is specified, assume it's the set of all real numbers.

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