[Math] Finding a general solution for {sin(x)=sqrt(3)/2 , cos(x)=-0.5} using an algebraic method

Question

I am trying to form a trig general solution for this system of equations:
$${ \sin(x)=\frac{\sqrt{3}}{2} , \cos(x)=-\frac{1}{2}}.$$

I found out that the general solution is $$x=2\pi n -\frac{4\pi}{3}\
$$ I did this by solving each equation seperately, finding the values of x that satisfy both, and then working out the general solution by looking at the pattern. However I want to find a way of doing it algebraically.

I tried using the identity $$\frac{\sin(x)}{\cos(x)}=\tan(x)$$ Giving me one equation:
$$\tan(x)=-\sqrt{3}.$$

Then I found the general solution for this equation $$x=\pi n-\frac{\pi }{3}$$
However this equation is not the correct general solution – whilst it does encompass all of the correct solutions, it also gives incorrect solutions (the incorrect solutions lie when even values of n are subbed in).

Can anybody help me find a foolproof method for finding a general solution for a pair of trig equations?

Thanks.

Answer 1

For this special case you can write:

$$z=\cos(x)+i\sin(x)=-\frac{1}{2}+i\frac{\sqrt{3}}{2}=e^{i(\frac{2}{3}\pi+2\pi k)}$$

Note that $\frac{2}{3}\pi+2\pi k=-\frac{4}{3}\pi+2\pi k'$.

You instantly get all solutions.