3 equations, 3 variables: can it be solved?

[azizz]

For a kinematic problem I obtained the following equations:

p1=- r*sin(theta_c) - d*cos(psi_a)*sin(theta_a)+c1
p2=d*sin(psi_a)*sin(theta_a)+c2
p3=d*cos(theta_a) + r*cos(theta_c)+c3

I want to solve these equations for theta_a, psi_a, theta_c, assuming that all other variables are known.
The solutions should be within these constraints:
-pi/4<psi_a<pi/4
pi/4<theta_a<3*4/pi
-pi/2<theta_c<pi/2

Can someone help me solving this problem? I think it should be possible, but it is quite hard...

Thx in advance

 

[micromass]

Here's how you can solve it (but it's probably going to be a lot of manipulation):

Transform every cosine in the system into a sine using the formula $$\cos\theta=\pm\sqrt{1-\sin^2\theta}$$ (plus or minus depending on in what quadrant the theta is).

Now introduce the following variables: $$\sin(\theta_a)=x,~\sin(\psi_a)=y,~\sin(\theta_c)=z$$. Now you have a system of equations in x, y and z. And it is not too hard to solve this, I think...

 

[azizz]

That works indeed.
Thanks a lot!