# Thread: Help to solve trigonometric equation for theta

1. ## Help to solve trigonometric equation for theta

Hi,

I'm looking for some help or guidance to solve the following equation for theta as a function of R2. L1 and L2 are constants and the specific range that I'm interested in for theta is 0 to pi/4.

2. ## Re: Help to solve trigonometric equation for theta

Mathematica says

$\Large \theta = \tan ^{-1}\left(\frac{-\sqrt{L_1^2 R_2^2-L_2^2 R_2^2+R_2^4}-L_1 L_2}{L_1^2+R_2^2},\frac{-\frac{L_2 L_1^2}{L_1^2+R_2^2}-\frac{L_1 \sqrt{-R_2^2 \left(-L_1^2+L_2^2-R_2^2\right)}}{L_1^2+R_2^2}+L_2}{R_2}\right)$

where the $\tan^{-1}(x,y)$ function takes two arguments and is quadrant aware.

3. ## Re: Help to solve trigonometric equation for theta

To solve an equation of the form

$\displaystyle r \sin \theta =a+b \cos \theta$

square both sides and let $\displaystyle c=\cos \theta$ to get a quadratic equation in $\displaystyle c$

$\displaystyle \left(b^2+r^2\right)c^2+2 a b c +\left(a^2-r^2\right) =0$

where $\displaystyle \frac{\sqrt{2}}{2}<c<1$

The discriminant must be positive so a necessary condition is that $\displaystyle r^2\geq a^2-b^2$