How do you find the solution to 6cos^2theta+5costheta-4=0 if 0<=theta<360?

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May 25, 2018

The solutions are $S = \left\{{60}^{\circ} , {300}^{\circ}\right\}$ for $\theta \in \left[0 , 360\right)$

Explanation:

This is a quadratic equation in $\cos \theta$

$6 {\cos}^{2} \theta + 5 \cos \theta - 4 = 0$

The discriminant is

$\Delta = {b}^{2} - 4 a c = {5}^{2} - 4 \left(6\right) \left(- 4\right) = 121$

As $\Delta > 0$, there are $2$ real solutions

$\left\{\begin{matrix}\cos {\theta}_{1} = \frac{- 5 + \sqrt{121}}{12} = \frac{- 5 + 11}{12} \\ \cos {\theta}_{2} = \frac{- 5 - \sqrt{121}}{12} = \frac{- 5 - 11}{12}\end{matrix}\right.$

$\left\{\begin{matrix}\cos {\theta}_{1} = \frac{6}{12} = \frac{1}{2} \\ \cos {\theta}_{2} = - 1.33\end{matrix}\right.$

{(theta_1=60^@ ;300^@),(theta_2=O/):}

The solutions are $S = \left\{{60}^{\circ} , {300}^{\circ}\right\}$ for $\theta \in \left[0 , 360\right)$

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