# solving for x trig functions

I am having trouble understanding the trigonometry of solving for "X". I got to this point from a problem but now what is left is this trig/algebra solving for "X" but I cant figure out how to get the "X".

Process: 100 + mg*3cos(u) - ky*6cos(u) =0 ;

y = 6 sin(u) ;

100 + 600*cos(X) - 50*6sin(X) * 6cos(X) =0;

cos(X) = 0.919208 , 0.0833777

Answers: X = 23 and 85 degrees

• Do you want exact solutions? If not, you can use the inverse trig functions. – Jean-François Gagnon Sep 29 '15 at 4:29
• No, I have the answers (in bold) but I don't know how to get to them. I am rusty with my trig functions. – Stan-Lee Sep 29 '15 at 4:41

You have to get it in terms of one function, not two. If you know that $\sin X \ge 0$, then you can substitute $\sqrt{1 + \cos^2 x}$ for $\sin x$. Then solve the resulting equation for $\cos x$ (move everything but the square root to the other side, then square both sides - be warned that this may introduce false solutions, so check all values against the unsquared equation). Once you know the possible values of $\cos x$, you can take the inverse cosine to find the values of $x$.
• Inverse cosine is the $cos^{-1}$ button on the calculator. – Jean-François Gagnon Sep 29 '15 at 13:35