Trigonometry solutions Question

[PhamCy]

Summary: https://www.physicsforums.com/threads/trigonometry-question.977263/

Here's the question.
Find the solutions of the equation tan(x)=2cos(x)+1 if 0 ≤ x ≤ 2π.
I know this question can be solved by observing the graph but is there any other ways (like algorithms OR some Trigonometry rules) to solve this, I couldn't figure that out.
Thank you for those who noticed my question.

[Moderator's note: Moved from a technical forum and thus no template.]

 

[BvU]

Hello PhamCy, !

Nice try, but PF asks a little more from you: make a start with the solution and we'll help you.

For example: substitute $\tan x = {\sin x\over \cos x}$ and see where it leads you

 

[mfb]

see where it leads you

Way beyond what you can solve analytically with high school mathematics. WolframAlpha knows how to solve it analytically, here is one of the solutions:

 

[BvU]

Oops, I 'missed' the +1 which makes things difficult

 

[vela]

You could express the relationship as a polynomial in $\cos\theta$, and then find the roots of the polynomial using something like Newton's method.

 

[hunt_mat]

Turn everything into $\cos x$ to get:
$$4\cos^{4}x+4\cos^{3}x+2\cos^{2}x-1=0$$
From here solve it using whatever method you like. I would advise you plotting the polynomial first.