Find all $\displaystyle x$ such that $\displaystyle 13\sin{x}-84\cos{x} = 17$.
EDIT: the (main) title is wrong, it should be 13sinx-84cosx = 17.
$\displaystyle A\sin{x} - B\cos{x} = C$
$\displaystyle A\sin{x} - B\cos{x} = R\sin(x - \alpha)$
$\displaystyle R = \sqrt{A^2+B^2}$
$\displaystyle \alpha = \arctan\left(\dfrac{B}{A}\right) = \arctan\left(\dfrac{84}{13}\right)$
so, assuming $\displaystyle 0 \le x < 2\pi$ ...
$\displaystyle 13\sin{x} - 84\cos{x} = 85\sin(x - \alpha)$
$\displaystyle 85\sin(x - \alpha) = 17
$
$\displaystyle \sin(x - \alpha) = \dfrac{1}{5}$
$\displaystyle x - \alpha = \arcsin\left(\dfrac{1}{5}\right)$
$\displaystyle x - \alpha = \pi - \arcsin\left(\dfrac{1}{5}\right)$
@Resilient
I had one just like this and I found a way to get the correct solution with pencil but... your's is a bit to nasty to get a clean radian solution as skeeter says. you can check my thread how I did almost the same problem.
http://www.mathhelpforum.com/math-he...on-172605.html
I got yours into quadratic form in sin function but the coefficients have no common factors other then 1
$\displaystyle 13sin\theta-84cos\theta=17 $
$\displaystyle (-84\sqrt{1-sin^2\theta})^2=(17-13sin\theta)^2 $
$\displaystyle 7225sin^2\theta-442sin\theta-6767=0 $
$\displaystyle (7225:5^2,17^2)(442:2,13,17)(6767:67,101)$