Solve the equation for θ if .

Question

Solve the equation for θ  if  .  ​

Quizplus · Accepted Answer

We notice that the given expression can be written as:

$$\frac{\sqrt{3} \sin	heta + \cos	heta}{\sin	heta + \sqrt{3}\cos	heta} = \frac{\sqrt{3}(\sin	heta + \sqrt{3}\cos	heta) + \cos	heta - \sqrt{3}\cos	heta}{(\sin	heta + \sqrt{3}\cos	heta) - \sqrt{3}(\sin	heta + \sqrt{3}\cos	heta)} $$

Simplifying this we get:

$$	an\left(	heta + \frac{\pi}{6}ight) = -\sqrt{3}$$

Thus, we get:

$$	heta + \frac{\pi}{6} = \frac{5\pi}{6}, \frac{11\pi}{6}$$

Which gives us:

$$	heta = \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{10\pi}{3}$$

Out of these, only $	heta = 0$ and $	heta = \frac{4\pi}{3}$ lie within the range $[0,2\pi)$. Thus, the answer is $\boxed{	extbf{(C) }0^\circ, 240^\circ}$.

Solve the Equation for θ If