Question

In Problems 57– 80, solve each equation on the interval $0\le \theta \le 2\mathrm{\pi }$

$2{\cos }^2\theta -7\cos \theta -4=0$

Short answer

The solution set of the given equation is$\left\{\frac{2\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}\right\}$

Step-by-step solution

Step 1. Given Information 

In the given problems we have to solve each equation on the interval $0\le \theta \le 2\mathrm{\pi }$

$2{\cos }^2\theta -7\cos \theta -4=0$

Step 2. The equation in its present form contains a cosine.

Factor the equation.

$$\begin{gathered} 2{\cos }^2\theta -(8-1)\cos \theta -4=0 \\ 2{\cos }^2\theta -8\cos \theta +\cos \theta -4=0 \\ 2\cos \theta (\cos \theta -4)+1(\cos \theta -4)=0 \\ (2\cos \theta +1)(\cos \theta -4)=0 \end{gathered}$$

Step 3. Use the Zero-Product Property. 

$$\begin{gathered} 2\cos \theta +1=0\mathrm{or}\cos \theta -4=0 \\ 2\cos \theta +1-1=0-1\mathrm{or}\cos \theta -4+4=0+4 \\ 2\cos \theta =-1\mathrm{or}\cos \theta =4 \\ \frac{2}{2}\cos \theta =-\frac{1}{2}\mathrm{or}\cos \theta =4 \\ \cos \theta =-\frac{1}{2}\mathrm{or}\cos \theta =4 \end{gathered}$$

Step 4. Solving each equation in the interval [0,2π], we obtain  

$\theta =\frac{2\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}\mathrm{or}\mathrm{No}\mathrm{solution}$

The solution set is$\left\{\frac{2\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}\right\}$