Question

give the smallest two solutions of $\cos(6\theta)=-0.0512$ on $0,2\pi)$. separate the two solutions with a comma.</font color>

Explanation:

Step1: Find the general solution

We know that if \(\cos\alpha = x\), then \(\alpha=\cos^{- 1}(x)+2\pi k\) or \(\alpha = 2\pi-\cos^{-1}(x)+2\pi k\), where \(k\in\mathbb{Z}\).

Given \(\cos(6\theta)=- 0.0512\), first find \(\cos^{-1}(-0.0512)\approx1.623\)

So \(6\theta=\cos^{-1}(-0.0512)+2\pi k\) or \(6\theta = 2\pi-\cos^{-1}(-0.0512)+2\pi k\)

Step2: Solve for \(\theta\) when \(k = 0\)

For \(6\theta=\cos^{-1}(-0.0512)+2\pi\times0\), \(\theta=\frac{\cos^{-1}(-0.0512)}{6}\approx\frac{1.623}{6}=0.2705\)

For \(6\theta=2\pi-\cos^{-1}(-0.0512)+2\pi\times0\), \(6\theta = 2\pi - 1.623\approx6.283 - 1.623=4.66\), \(\theta=\frac{4.66}{6}\approx0.777\)

Answer:

\(0.2705,0.777\)