Trigonometry question, 2 unknowns

Question

Given $\cos\theta=k$ and $\sin\theta=k\sqrt3$, where $k<0$ and $0<\theta<2\pi$, find $\theta$ and $k$.

I have no idea how to even begin. I have found that $\theta$ lies in the third quadrant. I have no idea how to continue. Any help please?

Answer 1

Hint: $ 1= \cos^2 \theta+ \sin^2 \theta= 4k^2$, hence $k= \pm \frac{1}{2}$.

Answer 2

Another solution would be to use the identity that $\frac{\sin\theta}{\cos\theta} = \tan\theta$. Hence in your case, we would get that $\tan\theta = \frac{k\sqrt{3}}{k}= \sqrt{3}$. To find $\theta$ take the inverse tan ($\arctan$) of $\sqrt{3}$ and then plug this answer back into your original equations involving $\sin$ and $\cos$. From there you can simply rearrange for $k$.

Hope this helps!