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Chapter 5: Problem 8

Verify that the \(x\) -values are solutions of the equation. \(2 \cos ^{2} 4 x-1=0\) (a) \(x=\frac{\pi}{16}\) (b) \(x=\frac{3 \pi}{16}\)

Short Answer

Expert verified
Yes, both \(x = \frac{\pi}{16}\) and \(x = \frac{3\pi}{16}\) are solutions of the given equation.

Step by step solution

01

Understand the equation

The given equation is in the form of a trigonometric function involving the cosine function. It is a simple equation of the type \(2 cos^2(ax) - 1 = 0\), where \(a\) is a constant.
02

Substituting \(x = \frac{\pi}{16}\)

The substitute \(x = \frac{\pi}{16}\) into the original equation yields \(2 cos^2(4 \times \frac{\pi}{16}) - 1\). This simplifies to \(2 cos^2(\frac{\pi}{4}) - 1\). The cosine of \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\), resulting in the expression \(2 (\frac{\sqrt{2}}{2})^2 - 1 = 0\). Simplifying this results in the equation 0 = 0, which is true and confirms that \(x = \frac{\pi}{16}\) is a solution for the equation.
03

Substituting \(x = \frac{3\pi}{16}\)

The substitute \(x = \frac{3\pi}{16}\) into the original equation yields \(2 cos^2(4 \times \frac{3\pi}{16}) - 1\). This simplifies to \(2 cos^2(\frac{3\pi}{4}) - 1\). The cosine of \(\frac{3\pi}{4}\) is \(-\frac{\sqrt{2}}{2}\), resulting in the expression \(2 (-\frac{\sqrt{2}}{2})^2 - 1 = 0\). Simplifying this results in the equation 0 = 0, which is true and confirms that \(x = \frac{3\pi}{16}\) is a solution for the equation.

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