Trigonometry
Trigonometry starts with the sides of a right triangle and ends with waves, vectors and polar curves. How does one idea stretch that far?
Where you are 7 inside
What this area is about
Trigonometry links angles to lengths. In a right triangle, the ratio of two sides depends only on an acute angle. Placing the angle on a circle of radius 1 extends sine and cosine to every real input, so they become periodic functions whose graphs are waves. From there come identities, triangles with no right angle, and new ways to describe direction and position.
How these fit together
Measuring angles. Angles and their measure lets an angle turn any amount in either direction and introduces radians.
Definitions of the functions. Right triangle trigonometry defines the six ratios for an acute angle; the unit circle extends them to angles of any size and sign, from the coordinates of a point on the circle.
Results and their uses. The laws of sines and cosines solve any triangle. Trigonometric functions and their graphs draw, transform and invert the waves. Trigonometric identities are equations true at every input where both sides are defined, and solve trig equations.
Applications. Polar coordinates and vectors describe direction and distance: vectors, parametric curves, polar graphs and the polar form of a complex number.
A path through it
Angles and their measure, and right triangle trigonometry, first; then the unit circle. After it, the laws of sines and cosines, and trig functions; trig identities follow trig functions. Polar coordinates and vectors come last, because they draw on all six.
Next, in calculus: the limit of \displaystyle \frac{\sin x}{x}, derivatives of trig functions and parametric, polar and vector valued functions.