This section has all been about calculating trigonometric functions and solving trigonometric equations. Trigonometry is a very practical branch of mathematics (if you're into that sort of thing).
It shows up in architecture, engineering, geography, astronomy, digital imaging, and a host of other fields. It can be used to calculate distances like the heights of mountains or how far away the stars in the sky are.
The cyclic, repeated nature of trig functions means that they are useful for studying different types of waves in nature: not just in the ocean, but the behavior of light, sound, and electricity as well. Another thing that repeats: pendulums. Yeah, you can make some cool stuff happen with trigonometry.
The Cartesian coordinate system is a fancy name for the graphs we've always made. They have two axes, x and y, and any point on the graph is defined by a coordinate with both an x and y component.
The Cartesian system doesn't have a monopoly on graphing, though. If it did, Teddy Roosevelt would come in and trust-bust it up one axis and across the other.
A different kind of coordinate system is the polar coordinate system. Instead of x and y, it defines a point using r, the radius, and θ (pronounced "theta"), for an angle.
Why might we want to use polar coordinates instead of our familiar Cartesian system? Compare the equations for the same circle, centered on the origin with radius 4:
Cartesian: x2 + y2 = 16
Polar: r = 4
We happen to think the polar equation is much easier to work with in this case.
There are a lot of interesting and cool to look at shapes that are much easier to define using polar coordinates. And because the coordinate system involves circles and angles so much, trigonometric equations are vital to working in this system.