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=(1)/(\cos \theta)`

Therefore, we have:

`\begin{gathered} (4)/(\cos\theta)-\sqrt[]{3}=\sqrt[]{3}+(7)/(\cos\theta) \\ \text{ We add }\sqrt[]{3}\text{ and subtract }(7)/(\cos\theta)\text{ from both sides to get:} \\ (4)/(\cos\theta)-(7)/(\cos\theta)=2\sqrt[]{3} \\ -(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:

`(150\pi)/(180)=(5\pi)/(6)`