Trigonometry! #89

Let $a = A\cos(\alpha)$ and $b = A\sin(\alpha)$ for some real $A,\alpha.$ Then

$c = a\cos(\theta) - b\sin(\theta) = A(\cos(\alpha)\cos(\theta) - \sin(\alpha)\sin(\theta)) = A\cos(\alpha + \theta).$

We also then have that

$a\sin(\theta) + b\cos(\theta) = A(\cos(\alpha)\sin(\theta) + \sin(\alpha)\cos(\theta)) =$

$A\sin(\alpha + \theta) = \pm A\sqrt{1 - \cos^{2}(\alpha + \theta))} = \pm A\sqrt{1 - (\frac{c}{A})^{2}} =$

$\pm \sqrt{A^{2} - c^{2}} = \boxed{\pm \sqrt{a^{2} + b^{2} - c^{2}}}$ ,

since $a^{2} + b^{2} = A^{2}(\sin^{2}(\alpha) + \cos^{2}(\alpha)) = A^{2}$ .

simple..

expn 1 is sqrt(a^2+b^2) cos( theta + arctan b/a ) = c

expn 2 is sqrt(a^2+b^2) sin( theta + arctan b/a ) = c

now just use sin^2 x+ cos^2 x = 1