# solving Trigonometric equations: $\textrm{cos}(5x)\textrm{cos}(x)=\textrm{sin}(5x)\textrm{sin}(x)-0.5$

Question: Find solutions for $\textrm{cos}(5x)\textrm{cos}(x)=\textrm{sin}(5x)\textrm{sin}(x)-0.5$ I did $\textrm{cos}(6x)=-1/2$ using the subtraction formula for cos. I'm confused how to find the solutions now since there are 12.

I thought you could just see that its $\cos$ of 20 degrees and see what solutions are in quadrant 1 and 4, but do you add the period to get the 12 solutions? If someone could explain how to find the twelve solutions that would be great.

• The equation can be rewritten as $\cos(6x)=\cos(2\pi/3)$. Then, use the general solution formula for the equation $\cos(a)=\cos(b)$ which is $a=2n\pi\pm b~\forall~n\in\Bbb Z$ to get the solution set. – learner Dec 16 '15 at 1:41

Hint: $\cos(6x)=\cos(5x)\cos(x)-\sin(5x)\sin(x)$
Think of the solutions to $\cos(x)=-\frac12$ and the fact that $\cos(x)$ has period $2\pi$.
• You have $\cos(6x)=-\frac12=\cos\left(\pm\frac{2\pi}{3}\right)$. Then remember that $\cos(x)=\cos(x\pm2\pi)=\cos(x\pm4\pi)$. – robjohn Dec 16 '15 at 1:42