Polar coordinates and vectors
Some quantities have a direction as well as a size, and some points are easier to find by distance and angle than by across and up. How does trigonometry describe them?
Where you are 10 inside
What this area is about
Three descriptions that x and y alone handle badly. A vector is a length with a direction, such as a velocity. Polar coordinates name a point by a distance and an angle, (r, \theta). Parametric equations give x and y each as a function of a parameter t, so a curve is traced in a definite direction.
How these fit together
Vectors. Vectors defines the arrow, its components \langle a, b \rangle and its length. Vector addition and scalar multiplication adds and scales. The dot product gives the angle between two nonzero vectors. Vectors in space carries these tools to \langle a, b, c \rangle, and the cross product returns a vector perpendicular to two space vectors, as long as the area of their parallelogram.
Curves traced in order. Parametric equations define the curve and its orientation; eliminating the parameter recovers one equation in x and y.
Polar. Polar coordinates plot and convert points and equations. Polar equations graph r = f(\theta). Polar form writes a nonzero complex number as r(\cos\theta + i\sin\theta), so multiplying multiplies lengths and adds angles.
A path through it
Vectors first: every other child needs it. Then three strands. Vector operations, the dot product, vectors in space, and last the cross product, which needs the dot product and vectors in space. Parametric equations, then eliminating the parameter. Polar coordinates, then polar equations and polar form.
Next come parametric, polar and vector valued functions in calculus and vectors in n dimensions in linear algebra.