I have stumbled into this equation
$sin (x) sin(12) - sin(24)sin(228-x) = 0$
all what I managed to do is to simplify it to
$sin(x)-2sin(228-x)cos12 = 0$
and all what I did after was useless
my question is how to solve this equation and what exactly should I learn to solve equations like these?
On
If they're looking for a numerical answer, the quickest way is (I think) to use a Weierstrass substitution. http://en.wikipedia.org/wiki/Tangent_half-angle_substitution. Mostly used in tricky integrals, but I found it handy here.
Let $\displaystyle x = \tan \frac{x}{2}$.
Then $\displaystyle \sin x = \frac{2t}{1+t^2}$ and $\displaystyle \cos x = \frac{1-t^2}{1+t^2}$.
Expand the $\displaystyle \sin(228-x)$ term using the angle sum formula and do the sub. You'll find it very quickly reduces to a quadratic equation in $\displaystyle t$ that can be solved with the general quadratic formula. After that take the arctangent and multiply by two to get $\displaystyle x$.
What did you do after? Did you use $sin(A+B)=sin(A)cos(B)+cos(A)sin(B)$? This should give you an expression for $tan(x)$ after a bit of simplification.