Solving trigonometric equation (decoupling)

I wanted to solve ϕ as a function of other variables

Is is possible to decouple this equation ?

Even matlab symbolic gave me some kind of log function. I don't understand what that mean actually.

any help is apperciated

Answer 1

[Alan Stevens]

How about just defining the function:

phi = @(psi, theta) atan( sin(psi).*sin(theta)./(cos(psi) + cos(theta)) );

[HN]

Alan Stevens , did you solve it by hand ? Why the matlab gives such complicated reuslt ?

If possible please give some tip how did you get it.

[HN]

Got it Alan Stevens ,

Thank you

[Alan Stevens]

Yes, I did it by hand; it was a fairly obvious solution! I've no idea why Matlab came up with a complicated solution.

If you add cos(theta)sin(phi) to both sides, then factor out the sin(phi) on the left hand side; then divide both sides by cos(phi) and both sides by cos(psi)+cos(theta), you have tan(phi) = ... .

[HN]

Thank you and apperciated !

Answer 2

[KSSV]

Hand calculation is also possible.....

syms phi psi theta 
eqn = cos(psi)*sin(phi)+cos(theta)*sin(phi)-cos(phi)*sin(psi)*sin(theta)==0
s = solve(eqn,psi)

[HN]

This is solution for phi

Actually ψ is given. But, same input to solve ϕ gave me something complex number like solution.

Did you try with phi ?

[HN]

This is for psi

Any tip for hand calculation ?

[KSSV]

[KSSV]

The same answer syms also gives with futher simplifications.

[HN]

Yeah, I just got it as soon as I see Alan Stevens 's answer.

First , I used matlab as I usually do along with many other equations and matlab's complicated result set my mind that this problem has complicatation.

And another funny experiance, a Paper with 400 over citation has mislead me in this problem. Eq.15 (in the paper) is what I've been trying to look at.

Thank you very much both of you !