In article <28131408.1130497985592.JavaMail.jaka...@nitrogen.mathforum.org>,
semiconductor <erh...@hotmail.com> wrote: >I have a question regarding the equations listed below. I want to simplify them as much as possible using trig identities so that I can solve for theta1, theta2, and theta3.
I hope you don't intend to work with these symbolically, since explicit expressions for the t_i will be ghastly. But perhaps you intend to do numerical computations to solve for t1,t2,t3 when all the other quantities are known numerically? In that case things aren't TOO bad. You have one equation that involves just one of the three variables:
This can be expanded to A*sin(t3) + B*cos(t3) = C where A,B,C do not involve t1, t2, or t3. Use sin(t) = 2*u/(1+u^2), cos(t) = (1-u^2)/(1+u^2) where u = tan(t3/2) to write your equation as a quadratic in u . Solve with quadratic formula and use arctan() to express t3 in terms of everything else.
Again I use use the half-angle substitutions (i.e., express in terms of u1 = tan(t1/2) and u2 = tan( (t1+t2)/2 ) ) and get a pair of biquadratic equations in u1 and u2. You'll have to solve a quartic polynomial this time: u2 is a root of