simultaneous equations with vector addition

Question

hi. I need help with solving these equations please for r, θ and φ

rcos(0)cos(307.8) = 648cos(φ)cos(θ) + 88cos(28)cos(242)

rcos(0)sin(307.8) = 648cos(φ)sin(θ) + 88 cos(28)sin(242)

rsin(0) = 648sin(φ)+ 88sin(28)

Answer 1

Since sin(0) = 0, phi = arcsin(-88/648sin(28)) =

Then cos(phi) = 0.998.

Then for the first and second equation, the only unknowns are r and theta.

So, subtract cos(307.8)/sin(307.8) times the second equation from the first equation to cancel out the left hand side.

Now subtract the sin(theta) term from both sides and then square both sides.

Now make the substitution of of sin^2(theta) = 1 - cos^2(theta) and collect terms on one side of the equation.

Then solve the quadratic equation for cos(theta).

Then take arccos() of both sides leaving the answer.