# Solving a system of trigonometric equations, for x and y (multivariable calculus).

#### Mik

Alright, here's the problem:

theta = arctan(y/(x-1)) and phi = arctan(y/(x+1))

I started off by taking the tangent of each side, which seemed obvious enough. So tan(theta) = y/x-1 and tan(phi) = y/x+1, but now I don't know of any method to isolate just the y and the x. Multiplying x-1 and x+1 to the other side doesn't help matters much, since it just adds an x to the other side that you still can't isolate.

Anybody know how to go about solving this? Thanks.

#### ILikeSerena

MHF Helper
Re: Solving a system of trigonometric equations, for x and y (multivariable calculus)

Alright, here's the problem:

theta = arctan(y/(x-1)) and phi = arctan(y/(x+1))

I started off by taking the tangent of each side, which seemed obvious enough. So tan(theta) = y/x-1 and tan(phi) = y/x+1, but now I don't know of any method to isolate just the y and the x. Multiplying x-1 and x+1 to the other side doesn't help matters much, since it just adds an x to the other side that you still can't isolate.

Anybody know how to go about solving this? Thanks.
Hi Mik!

You have the equations:
tan(theta) = y/(x-1) and tan(phi) = y/(x+1)

(x-1)tan(theta) = y and (x+1)tan(phi) = y​

Equate the equations to each other:
(x-1)tan(theta) = (x+1)tan(phi)​

Solve for x...
Then calculate y from y=(x+1)tan(phi).