Question

What are the values of `theta` that satisfies `2costhetatantheta + 2costheta -1 = tantheta`?

Answer 1

To summarize, solutions are:
•`theta = pi/3 + 2pin`
•`theta = (5pi)/3 + 2pin`
•`theta = pin`

Explanation:

We start by putting all terms to one side:

`2costhetatantheta - tan theta + 2costheta - 1 = 0`

`tantheta(2costheta - 1) + 2costheta - 1 = 0`

`(2costheta - 1)tan theta = 0`

`costheta = 1/2 or tan theta = 0`

If we consider the first equation, we know that cosine will be positive in the first and forth quadrants. The reference angle will be `pi/3`, so the two solutions are going to be `pi/3` and `2pi - pi/3 = (5pi)/3`. The period of the cosine function is `2pi`, thus, the general form solution are :

`theta = pi/3 + 2pin and (5pi)/3 + 2pin`

For the second equation, we know that `tantheta = sintheta/costheta`, so the equation will be true only when `sintheta = 0`. This will occur when `theta = 0 or pi`. Since tangent has a period of `pi`, our general form solutions will be

`theta = pin`

Hopefully this helps!