system
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human
My TI-84 can perform a variety of mathematical operations, but cannot find the intersection points of two polar curves. Why?

This was an issue that was brought up in my AP Calculus class, and my teacher was hard-pressed to find an answer to the *why*.  She hazarded a guess that it had something to do with the (r, theta) = (r, theta + 2kpi) and (r, theta) = (-r, theta + pi + 2kpi) conditions when solving for intersection points, but we never really got to the bottom of it.

So, how/why is a calculator limited in this respect, and what other similar limitations do they have?
gpt
It's not "a calculator" limitation.

Polar coordinates are implemented as raw parametrics with one variable being modular with period 2 Pi. That's tricky to solve. You have to solve (ignoring that you can't do % on floats) 

    cos(theta1)*radius(theta1) = cos(theta2)*radius(theta2) 

and 

    sin(theta1)*radius(theta1) = sin(theta2)*radius(theta2) 

For instance, you can't just solve for theta1=theta2, because the intersection could be at Pi/4, 5 Pi/4 with r1 = - r2.

Basically, the TI 84 solves xy equations through newton's method. You're asking it to solve something not solvable by that.

Remember that with polars where you have r(theta), you actually just have 

x = r(theta) * cos(theta)

y = r(theta) * sin(theta)

However, "calculators" are not limited by that, just certain calculators.

You want a CAS = computer aided solver, such as the TI 89, or hell, Mathematica (Wolfram Alpha's math engine)

TL;DR: Solving intersection of y1 = f(x) and y2 = g(x) can be numerically solved over one variable via newton's method. Polar equations are really just parametric equations, which require solving over two variables.