Question

Solve the equation for all solutions in the interval. Write each answer in radians.

Answer 1

We are given a problem regarding trigonometric identities and asked to resolve for theta.

$4\sec \theta-\sqrt[]{3}=\sqrt[]{3}+7\sec \theta$

Recall that:

$\sec \theta=\frac{1}{\cos \theta}$

Therefore, we have:

$\begin{gathered} \frac{4}{\cos\theta}-\sqrt[]{3}=\sqrt[]{3}+\frac{7}{\cos\theta} \\ \text{ We add }\sqrt[]{3}\text{ and subtract }\frac{7}{\cos\theta}\text{ from both sides to get:} \\ \frac{4}{\cos\theta}-\frac{7}{\cos\theta}=2\sqrt[]{3} \\ -\frac{3}{\cos\theta}=2\sqrt[]{3} \\ \text{Multiply both sides by }\cos \theta\text{ and divide both sides by }2\sqrt[]{3}\text{ to get:} \\ \cos \theta=-\frac{3}{2\sqrt[]{3}}=-\frac{\sqrt[]{3}}{2} \\ \theta=\cos ^{-1}(-\frac{\sqrt[]{3}}{2})=150^o \\ \end{gathered}$

In radians, we have:

$\frac{150\pi}{180}=\frac{5\pi}{6}$