Question

give the smallest two solutions of $\cos(5\theta)=-0.7833$ on $0,2\pi)$. separate the two solutions with a comma.

Explanation:

Step1: Find the reference angle

We know that if \(\cos(x)=a\), then \(x = \cos^{-1}(a)+ 2k\pi\) or \(x=2\pi-\cos^{-1}(a)+2k\pi\), where \(k\in\mathbb{Z}\).
Given \(\cos(5\theta)=- 0.7833\), first find \(\alpha=\cos^{-1}(0.7833)\). Using a calculator, \(\alpha\approx0.707\) radians.

Step2: Find the general solutions for \(5\theta\)

Since \(\cos(5\theta)=-0.7833\), the solutions for \(5\theta\) are:
\(5\theta=\cos^{-1}(-0.7833)+2k\pi\) or \(5\theta = 2\pi-\cos^{-1}(-0.7833)+2k\pi\)
We know that \(\cos^{-1}(-0.7833)=\pi - 0.707\approx2.4346\) and \(2\pi-\cos^{-1}(-0.7833)=2\pi-( \pi - 0.707)=\pi + 0.707\approx3.8492\)
So \(5\theta\approx2.4346 + 2k\pi\) or \(5\theta\approx3.8492+2k\pi\)

Step3: Solve for \(\theta\)

For \(5\theta\approx2.4346 + 2k\pi\), \(\theta=\frac{2.4346+2k\pi}{5}\)
When \(k = 0\), \(\theta_1=\frac{2.4346}{5}=0.4869\)
For \(5\theta\approx3.8492+2k\pi\), \(\theta=\frac{3.8492 + 2k\pi}{5}\)
When \(k = 0\), \(\theta_2=\frac{3.8492}{5}=0.7698\)

Answer:

\(0.4869,0.7698\)