Question

solve ( 8 cos ( 5 x ) = 2 ) for the smallest three positive solutions. give your answers accurate to at least two decimal places, as a list separated by commas.

Explanation:

Step1: Isolate the cosine function

Divide both sides of the equation \(8\cos(5x) = 2\) by \(8\).
\(\cos(5x)=\frac{2}{8}=\frac{1}{4}\)

Step2: Use the inverse cosine function

If \(\cos\theta = a\), then \(\theta=\cos^{- 1}(a)+2k\pi\) or \(\theta = 2\pi-\cos^{-1}(a)+2k\pi\), where \(k\in\mathbb{Z}\). Here \(\theta = 5x\) and \(a=\frac{1}{4}\).
First, find \(5x\):
\(5x=\cos^{-1}(\frac{1}{4})+2k\pi\) or \(5x = 2\pi-\cos^{-1}(\frac{1}{4})+2k\pi\)

Step3: Solve for \(x\)

For \(5x=\cos^{-1}(\frac{1}{4})+2k\pi\), \(x=\frac{\cos^{-1}(\frac{1}{4})+2k\pi}{5}\).
For \(5x = 2\pi-\cos^{-1}(\frac{1}{4})+2k\pi\), \(x=\frac{2\pi-\cos^{-1}(\frac{1}{4})+2k\pi}{5}\).

We know that \(\cos^{-1}(\frac{1}{4})\approx1.3181\) (in radians).

When \(k = 0\) for \(x=\frac{\cos^{-1}(\frac{1}{4})+2k\pi}{5}\):
\(x_1=\frac{1.3181}{5}\approx0.26\)

When \(k = 0\) for \(x=\frac{2\pi-\cos^{-1}(\frac{1}{4})+2k\pi}{5}\):
\(x_2=\frac{2\pi - 1.3181}{5}=\frac{6.2832 - 1.3181}{5}=\frac{4.9651}{5}\approx0.99\)

When \(k = 1\) for \(x=\frac{\cos^{-1}(\frac{1}{4})+2k\pi}{5}\):
\(x_3=\frac{1.3181+2\pi}{5}=\frac{1.3181 + 6.2832}{5}=\frac{7.6013}{5}\approx1.52\)

Answer:

\(0.26,0.99,1.52\)