Help on simplifying trigonometric equations

After certain operations, i end up with 2 equations,

k1*sin(c1)+k2*sin(c2-c1) = l1*cos(x1) - l2*cos(x1+x2)
k2*cos(c2-c1)-k1*cos(c1) = l1*sin(x1) - l2*sin(x1+x2)

I need to get eqations for x1 and x2 , is there a way to get this simplified using MATLAB?

Answer 1

[Walter Roberson]

syms c1 c2 k1 k2 l1 l2 x1 x2
eqns = [k1*sin(c1)+k2*sin(c2-c1) == l1*cos(x1) - l2*cos(x1+x2)
        k2*cos(c2-c1)-k1*cos(c1) == l1*sin(x1) - l2*sin(x1+x2)]
sol = solve(eqns, [x1 x2])

This may take some time when it is first done -- about 90 seconds.

The result sol.x1 is two atan() expressions that differ in sign at a single location about 1600 characters into the 4000-ish character expression.

The result sol.x2 is two atan() expression that differ in leading sign (that is, one is the negative of the other)

In the form of solve() above you will get a warning

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'

In practice, if you ask for return conditions, it will think for a fair while... and then tell you that it was not able to come up with any solutions at all.

That should be taken as a hint to verify the results numerically a couple of times before relying on the solution; it might be the case that these are false roots.

[Adithya Vimukthi]

I run this and got output as,

sol = 

    x1: [0x1 sym]
    x2: [0x1 sym]

but i need simplifyed equantions for x1 and x2. Is there a way to get?

[Walter Roberson]

Which release are you using? With R2022b I get

sol.x1(1) = str2sym(horzcat('-2*atan((((8*(k1*l1^2*l2 - k2*l1^2*l2 - 2*k2*l1^2*l2*tan(c1/2)^2 - k1*l1^2*l2*ta', ...
'n(c1/2)^4 - k2*l1^2*l2*tan(c1/2)^4 + 2*k1*l1^2*l2*tan(c1/2 - c2/2)^2 + k1*l1^2*l', ...
'2*tan(c1/2 - c2/2)^4 + k2*l1^2*l2*tan(c1/2 - c2/2)^4 - 2*k1*l1^2*l2*tan(c1/2)^4*', ...
'tan(c1/2 - c2/2)^2 + 2*k2*l1^2*l2*tan(c1/2)^2*tan(c1/2 - c2/2)^4 - k1*l1^2*l2*ta', ...
'n(c1/2)^4*tan(c1/2 - c2/2)^4 + k2*l1^2*l2*tan(c1/2)^4*tan(c1/2 - c2/2)^4))/((2*k', ...
'1*k2 + 2*l1*l2 - k1^2*tan(c1/2)^2 - k2^2*tan(c1/2)^2 + l1^2*tan(c1/2)^2 + l2^2*t', ...
'an(c1/2)^2 - k1^2 - k2^2 + l1^2 + l2^2 - k1^2*tan(c1/2 - c2/2)^2 - k2^2*tan(c1/2', ...
' - c2/2)^2 + l1^2*tan(c1/2 - c2/2)^2 + l2^2*tan(c1/2 - c2/2)^2 - k1^2*tan(c1/2)^', ...
'2*tan(c1/2 - c2/2)^2 - k2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2)^2*ta', ...
'n(c1/2 - c2/2)^2 + l2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 2*k1*k2*tan(c1/2)^2 + 2', ...
'*l1*l2*tan(c1/2)^2 - 2*k1*k2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2 - c2/2)^2 + 2', ...
'*k1*k2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 +', ...
' 8*k1*k2*tan(c1/2)*tan(c1/2 - c2/2))*(k1^2*tan(c1/2)^2 - 2*k1*k2 + k2^2*tan(c1/2', ...
')^2 + l1^2*tan(c1/2)^2 - l2^2*tan(c1/2)^2 + k1^2 + k2^2 + l1^2 - l2^2 + k1^2*tan', ...
'(c1/2 - c2/2)^2 + k2^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c', ...
'1/2 - c2/2)^2 + 4*k1*l1*tan(c1/2) + k1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + k2^2*t', ...
'an(c1/2)^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c', ...
'1/2)^2*tan(c1/2 - c2/2)^2 - 4*k2*l1*tan(c1/2 - c2/2) + 2*k1*k2*tan(c1/2)^2 + 2*k', ...
'1*k2*tan(c1/2 - c2/2)^2 + 4*k1*l1*tan(c1/2)*tan(c1/2 - c2/2)^2 - 4*k2*l1*tan(c1/', ...
'2)^2*tan(c1/2 - c2/2) - 2*k1*k2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 8*k1*k2*tan(c1/', ...
'2)*tan(c1/2 - c2/2))) + (4*((2*l1*l2 - 2*k1*k2 + k1^2*tan(c1/2)^2 + k2^2*tan(c1/', ...
'2)^2 - l1^2*tan(c1/2)^2 - l2^2*tan(c1/2)^2 + k1^2 + k2^2 - l1^2 - l2^2 + k1^2*ta', ...
'n(c1/2 - c2/2)^2 + k2^2*tan(c1/2 - c2/2)^2 - l1^2*tan(c1/2 - c2/2)^2 - l2^2*tan(', ...
'c1/2 - c2/2)^2 + k1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + k2^2*tan(c1/2)^2*tan(c1/2', ...
' - c2/2)^2 - l1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c1/2)^2*tan(c1/2 - c', ...
'2/2)^2 + 2*k1*k2*tan(c1/2)^2 + 2*l1*l2*tan(c1/2)^2 + 2*k1*k2*tan(c1/2 - c2/2)^2 ', ...
'+ 2*l1*l2*tan(c1/2 - c2/2)^2 - 2*k1*k2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 2*l1*l2*', ...
'tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 8*k1*k2*tan(c1/2)*tan(c1/2 - c2/2))*(2*k1*k2 + ', ...
'2*l1*l2 - k1^2*tan(c1/2)^2 - k2^2*tan(c1/2)^2 + l1^2*tan(c1/2)^2 + l2^2*tan(c1/2', ...
')^2 - k1^2 - k2^2 + l1^2 + l2^2 - k1^2*tan(c1/2 - c2/2)^2 - k2^2*tan(c1/2 - c2/2', ...
')^2 + l1^2*tan(c1/2 - c2/2)^2 + l2^2*tan(c1/2 - c2/2)^2 - k1^2*tan(c1/2)^2*tan(c', ...
'1/2 - c2/2)^2 - k2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2)^2*tan(c1/2 ', ...
'- c2/2)^2 + l2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 2*k1*k2*tan(c1/2)^2 + 2*l1*l2*', ...
'tan(c1/2)^2 - 2*k1*k2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2 - c2/2)^2 + 2*k1*k2*', ...
'tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 8*k1*k', ...
'2*tan(c1/2)*tan(c1/2 - c2/2)))^(1/2)*(l1*l2 + l1*l2*tan(c1/2)^2 + l1*l2*tan(c1/2', ...
' - c2/2)^2 + l1*l2*tan(c1/2)^2*tan(c1/2 - c2/2)^2))/((2*k1*k2 + 2*l1*l2 - k1^2*t', ...
'an(c1/2)^2 - k2^2*tan(c1/2)^2 + l1^2*tan(c1/2)^2 + l2^2*tan(c1/2)^2 - k1^2 - k2^', ...
'2 + l1^2 + l2^2 - k1^2*tan(c1/2 - c2/2)^2 - k2^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c', ...
'1/2 - c2/2)^2 + l2^2*tan(c1/2 - c2/2)^2 - k1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - ', ...
'k2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l2^2', ...
'*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 2*k1*k2*tan(c1/2)^2 + 2*l1*l2*tan(c1/2)^2 - 2*', ...
'k1*k2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2 - c2/2)^2 + 2*k1*k2*tan(c1/2)^2*tan(', ...
'c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 8*k1*k2*tan(c1/2)*tan(', ...
'c1/2 - c2/2))*(k1^2*tan(c1/2)^2 - 2*k1*k2 + k2^2*tan(c1/2)^2 + l1^2*tan(c1/2)^2 ', ...
'- l2^2*tan(c1/2)^2 + k1^2 + k2^2 + l1^2 - l2^2 + k1^2*tan(c1/2 - c2/2)^2 + k2^2*', ...
'tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c1/2 - c2/2)^2 + 4*k1*l1', ...
'*tan(c1/2) + k1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + k2^2*tan(c1/2)^2*tan(c1/2 - c', ...
'2/2)^2 + l1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c1/2)^2*tan(c1/2 - c2/2)', ...
'^2 - 4*k2*l1*tan(c1/2 - c2/2) + 2*k1*k2*tan(c1/2)^2 + 2*k1*k2*tan(c1/2 - c2/2)^2', ...
' + 4*k1*l1*tan(c1/2)*tan(c1/2 - c2/2)^2 - 4*k2*l1*tan(c1/2)^2*tan(c1/2 - c2/2) -', ...
' 2*k1*k2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 8*k1*k2*tan(c1/2)*tan(c1/2 - c2/2))))*', ...
'(2*k1*k2 + 2*l1*l2 - k1^2*tan(c1/2)^2 - k2^2*tan(c1/2)^2 + l1^2*tan(c1/2)^2 + l2', ...
'^2*tan(c1/2)^2 - k1^2 - k2^2 + l1^2 + l2^2 - k1^2*tan(c1/2 - c2/2)^2 - k2^2*tan(', ...
'c1/2 - c2/2)^2 + l1^2*tan(c1/2 - c2/2)^2 + l2^2*tan(c1/2 - c2/2)^2 - k1^2*tan(c1', ...
'/2)^2*tan(c1/2 - c2/2)^2 - k2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2)^', ...
'2*tan(c1/2 - c2/2)^2 + l2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 2*k1*k2*tan(c1/2)^2', ...
' + 2*l1*l2*tan(c1/2)^2 - 2*k1*k2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2 - c2/2)^2', ...
' + 2*k1*k2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2)^2*tan(c1/2 - c2/2)', ...
'^2 + 8*k1*k2*tan(c1/2)*tan(c1/2 - c2/2)))/(4*(l1*l2 + l1*l2*tan(c1/2)^2 + l1*l2*', ...
'tan(c1/2 - c2/2)^2 + l1*l2*tan(c1/2)^2*tan(c1/2 - c2/2)^2)))'));

sol.x2(1) = str2sym(horzcat( '-2*atan(((2*l1*l2 - 2*k1*k2 + k1^2*tan(c1/2)^2 + k2^2*tan(c1/2)^2 - l1^2*tan(c1/', ...
'2)^2 - l2^2*tan(c1/2)^2 + k1^2 + k2^2 - l1^2 - l2^2 + k1^2*tan(c1/2 - c2/2)^2 + ', ...
'k2^2*tan(c1/2 - c2/2)^2 - l1^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c1/2 - c2/2)^2 + k1', ...
'^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + k2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - l1^2*t', ...
'an(c1/2)^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 2*k1*k2*ta', ...
'n(c1/2)^2 + 2*l1*l2*tan(c1/2)^2 + 2*k1*k2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2 ', ...
'- c2/2)^2 - 2*k1*k2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2)^2*tan(c1/', ...
'2 - c2/2)^2 - 8*k1*k2*tan(c1/2)*tan(c1/2 - c2/2))*(2*k1*k2 + 2*l1*l2 - k1^2*tan(', ...
'c1/2)^2 - k2^2*tan(c1/2)^2 + l1^2*tan(c1/2)^2 + l2^2*tan(c1/2)^2 - k1^2 - k2^2 +', ...
' l1^2 + l2^2 - k1^2*tan(c1/2 - c2/2)^2 - k2^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2', ...
' - c2/2)^2 + l2^2*tan(c1/2 - c2/2)^2 - k1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - k2^', ...
'2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l2^2*ta', ...
'n(c1/2)^2*tan(c1/2 - c2/2)^2 - 2*k1*k2*tan(c1/2)^2 + 2*l1*l2*tan(c1/2)^2 - 2*k1*', ...
'k2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2 - c2/2)^2 + 2*k1*k2*tan(c1/2)^2*tan(c1/', ...
'2 - c2/2)^2 + 2*l1*l2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 8*k1*k2*tan(c1/2)*tan(c1/', ...
'2 - c2/2)))^(1/2)/(2*k1*k2 + 2*l1*l2 - k1^2*tan(c1/2)^2 - k2^2*tan(c1/2)^2 + l1^', ...
'2*tan(c1/2)^2 + l2^2*tan(c1/2)^2 - k1^2 - k2^2 + l1^2 + l2^2 - k1^2*tan(c1/2 - c', ...
'2/2)^2 - k2^2*tan(c1/2 - c2/2)^2 + l1^2*tan(c1/2 - c2/2)^2 + l2^2*tan(c1/2 - c2/', ...
'2)^2 - k1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - k2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2', ...
' + l1^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + l2^2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 - 2', ...
'*k1*k2*tan(c1/2)^2 + 2*l1*l2*tan(c1/2)^2 - 2*k1*k2*tan(c1/2 - c2/2)^2 + 2*l1*l2*', ...
'tan(c1/2 - c2/2)^2 + 2*k1*k2*tan(c1/2)^2*tan(c1/2 - c2/2)^2 + 2*l1*l2*tan(c1/2)^', ...
'2*tan(c1/2 - c2/2)^2 + 8*k1*k2*tan(c1/2)*tan(c1/2 - c2/2)))' ));

sol

sol = struct with fields:
    x1: -2*atan((((8*k1*l1^2*l2 - 8*k2*l1^2*l2 - 16*k2*l1^2*l2*tan(c1/2)^2 - 8*k1*l1^2*l2*tan(c1/2)^4 - 8*k2*l1^2*l2*tan(c1/2)^4 + 16*k1*l1^2*l2*tan(c1/2 - c2/2)^2 + 8*k1*l1^2*l2*tan(c1/2 - c2/2)^4 + 8*k2*l1^2*l2*tan(c1/2 - c2/2)^4 - 16*k1*l1^2*l2*tan…
    x2: -2*atan(((2*l1*l2 - 2*k1*k2 + k1^2*tan(c1/2)^2 + k2^2*tan(c1/2)^2 - l1^2*tan(c1/2)^2 - l2^2*tan(c1/2)^2 + k1^2 + k2^2 - l1^2 - l2^2 + k1^2*tan(c1/2 - c2/2)^2 + k2^2*tan(c1/2 - c2/2)^2 - l1^2*tan(c1/2 - c2/2)^2 - l2^2*tan(c1/2 - c2/2)^2 + k1^2*…

Ah, looks like you can potentially reduce sizes a fair bit

x1_better = simplify(sol.x1, 'steps', 50);
x2_better = simplify(sol.x2, 'steps', 50);

x1_better

x1_better = 

x2_better

x2_better =