A trigonometric equation

tan(x + y) = tanx · tan y

 

 

 

 

The equation tan(x + y) = tanx * tan y corresponds to the graph in figure 1. It should really consists of closed simple curves, the gaps is due to the computer program. The “bubbles” are symmetric around the line… y = x, they are infinite in number and separated by π in x- and y-direction. This is naturally a direct consequence of the period length π of the tangent function. The second picture is a magnification of the “bubble” going through the origin, in sequel this “bubble” is called B0. The third picture shows tan x versus tan y.

A few challenges:

1. Find some “simple” solution (e g rational multiple of π or maybe a little less simple) corresponding to a point on B0.

2. Solve the equation for y, to get y = f(x) for intervals corresponding to a points on B0.

3. Find the exact coordinates for the point where B0 intersects the line y = x.