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 $\displaystyle r^2\geq a^2-b^2$