Parametric Equations - Finding Tangent Lines

[r34racer01]

So I'm looking at this example in my textbook where they're trying to find the vert. & horizontal tangent lines of r = sin Θ, 0 < Θ < pi. they say to first change it to parametric equations where

x = r cos Θ = sin Θ cos Θ
and
y = r sin Θ = sin Θ sin Θ = sin^2 Θ

But now I'm just really confused, how did they get these parametric equations?

 

[haytil]

It's a simple conversion from polar to cartesian coordinates.

The definition of the conversion (I'm sure this is in your textbook) is:

x = r * cos Θ
y = r * sin Θ

However, you're given r as a function of Θ - i.e., r(Θ).

So the conversion becomes:

x = r(Θ) * cos Θ
y = r(Θ) * sin Θ

Simply replace r(Θ) with its definition (i.e., r(Θ) = sin Θ)

x = sin Θ * sin Θ
y = sin Θ * cos Θ