Question

give the smallest two solutions of $\cos(7\theta)=-0.1289$ on $0,2\pi)$. separate the two solutions with a comma.</font color>

Explanation:

Step1: Find the principal value

We know that if \(\cos\alpha = x\), then \(\alpha=\cos^{- 1}(x)\). Given \(\cos(7\theta)=-0.1289\), the principal value \(\alpha_1=\cos^{-1}(-0.1289)\approx1.6901\) (using a calculator in radian mode).

Step2: Use the property of the cosine function \(\cos\alpha=\cos(2\pi - \alpha)\)

The second - quadrant solution for \(\cos\alpha=-0.1289\) is \(\alpha_2 = 2\pi-\cos^{-1}(-0.1289)\approx2\pi - 1.6901=4.5931\)

Step3: Solve for \(\theta\)

Since \(7\theta=\alpha_1 + 2k\pi\) and \(7\theta=\alpha_2+2k\pi\) (\(k\in\mathbb{Z}\)).

For \(k = 0\):
When \(7\theta=\alpha_1\approx1.6901\), then \(\theta=\frac{1.6901}{7}\approx0.2414\)
When \(7\theta=\alpha_2\approx4.5931\), then \(\theta=\frac{4.5931}{7}\approx0.6562\)

Answer:

\(0.2414,0.6562\)