How do I do this fminsearch problem?

x^2 + 2*cos(θ) + y = α

x^2 −(3/2)*sin(θ) − y = 2.

Use the fminsearch function and the above system to find the minimum solution for α. Use initial guesses for x = 0 and y = 0. (Hint: This means solve the above equations by eliminating θ)

Answer 1

[Walter Roberson]

If

x^2 −(3/2)*sin(θ) − y = 2

then

x^2 −(3/2)*sin(θ) − y - x^2 + y = 2 - x^2 + y

so

-(3/2)*sin(theta) = 2 - x^2 + y

so

sin(theta) = 2/3 * (-2 + x^2 - y)

and then

theta = arcsine(2/3 * (x^2 - y - 2))

Substitute that into the first equation

x^2 + 2*cos( asin(2/3 * (x^2 - y - 2))) + y = alpha

and to minimize alpha, minimize the left hand side of that.