Solving four trigonometric equations, four unknowns

Please I want to solve these four trigonometric equations, four unknowns using a matlab code

10 cos ∅3 + 10 cos ∅2 − 𝑟1 = 0

15 cos ∅4 + 12 cos ∅2 − 𝑟1 = 0

10 sin∅3 + 10 sin∅2 = 0

15 sin∅4 + 12 sin∅2 = 0

thank you in advance.

Answer 1

[Torsten]

It seems that only 4 complex solutions exist.

syms phi2 phi3 phi4 r1 
eqn1 = 10*cos(phi3)+10*cos(phi2)-r1 == 0;
eqn2 = 15*cos(phi4)+12*cos(phi2)-r1 == 0;
eqn3 = 10*sin(phi3)+10*sin(phi2) == 0;
eqn4 = 15*sin(phi4)+12*sin(phi2) == 0;
sol = solve([eqn1,eqn2,eqn3,eqn4],[phi2 phi3 phi4 r1])

sol = struct with fields:
    phi2: [4×1 sym]
    phi3: [4×1 sym]
    phi4: [4×1 sym]
      r1: [4×1 sym]

double(sol.phi2)

ans = 
   1.5708 + 0.8858i
  -1.5708 - 0.8858i
   1.5708 - 0.8858i
  -1.5708 + 0.8858i

double(sol.phi3)

ans = 
  -1.5708 - 0.8858i
   1.5708 + 0.8858i
  -1.5708 + 0.8858i
   1.5708 - 0.8858i

double(sol.phi4)

ans = 
  -1.5708 - 0.5138i
   1.5708 + 0.5138i
  -1.5708 + 0.5138i
   1.5708 - 0.5138i

double(sol.r1)

ans = 
   0.0000 -20.1246i
   0.0000 -20.1246i
   0.0000 +20.1246i
   0.0000 +20.1246i

Answer 2

[Dyuman Joshi]

You can solve symbolically -

syms r1 t2 t3 t4
e1 = 10*cos(t3) + 10*cos(t2) - r1 == 0;
e2 = 15*cos(t4) + 12*cos(t2) - r1 == 0;
e3 = 10*sin(t3) + 10*sin(t2) == 0;
e4 = 15*sin(t4) + 12*sin(t2) == 0;
y=solve([e1 e2 e3 e4],[r1 t2 t3 t4])

y = struct with fields:
    r1: [4×1 sym]
    t2: [4×1 sym]
    t3: [4×1 sym]
    t4: [4×1 sym]

y.r1

ans = 

y.t2

ans = 

Use double() to get numeric values