Trigonometric functions and their graphs
Once an angle can be any real number, sine and cosine become functions with graphs. What do those graphs look like, and what can they model?
Where you are 8 inside
What this area is about
With the angle measured in radians, sine and cosine take any real input and give an output in [-1, 1], repeating every 2\pi. Their graphs are waves. Stretched and shifted, a wave fits anything that rises and falls on a steady cycle; cut down to one slice, it can run backward.
How these fit together
Definitions. Periodic functions defines a period and a cycle for any function.
The graphs. Graphing sine and cosine builds y = \sin x and y = \cos x from the unit circle. Graphing tangent and the reciprocal functions draws the four functions that divide by a coordinate, with vertical asymptotes where that coordinate is 0.
Changing the wave. Amplitude, period and phase shift reads each number of y = a\sin\big(b(x - h)\big) + k, with a \ne 0 and b \ne 0, off the formula. Writing an equation from a wave runs that backward, and modeling things that repeat builds the formula from facts about a wheel, a tide or a spring.
Inverses. Inverse trig functions keeps one slice of each graph so arcsine, arccosine and arctangent exist; composing trig and inverse trig functions evaluates expressions like \displaystyle \cos\left(\arcsin\frac{5}{13}\right) exactly.
A path through it
Two strands. In one, graphing sine and cosine comes first. After it, the tangent family, and separately amplitude, period and phase, then writing an equation from a wave, then modeling. In the other, periodic functions, then inverse trig functions, then compositions.
The inverse functions lead to solving basic trig equations and, in calculus, their derivatives.