Periodic Functions

Each day the sun rises in an easterly direction, approaches some maximum height, and sets in the west — and whatever the exact path a given location observes, it repeats, predictably, over and over. That is the shape this chapter studies: a periodic function, one whose values cycle on a fixed interval. Chapter 5 defined sine and cosine as coordinates of a point moving around the unit circle; graphing those values against the angle itself turns the circle’s endless revolution into a wave, whose height, cycle length, and horizontal position can each be stretched or shifted to model tides, temperatures, and anything else that oscillates. The remaining four trigonometric functions inherit graphs of their own, shaped by the vertical asymptotes their quotient definitions create. Finally, restricting each function to one well-chosen piece makes it invertible, and the inverse trigonometric functions answer the reverse question — given a ratio, what angle produced it? — the key to solving the equations of the next chapter.

Sections

• Graphs of the Sine and Cosine Functions — the sinusoidal graphs of $y=\sin x$ and $y=\cos x$, their amplitude, period, midline, phase shift, and vertical shift under the general forms $y=A\sin(Bx-C)+D$ and $y=A\cos(Bx-C)+D$, writing an equation from a graph, and modeling periodic behavior.
• Graphs of the Other Trigonometric Functions — the graphs of the tangent, cotangent, secant, and cosecant functions, their periods and vertical asymptotes, stretched and shifted variations of each, and reading a formula off a given graph.
• Inverse Trigonometric Functions — restricting the domains of sine, cosine, and tangent to define $\sin^{-1}x$, $\cos^{-1}x$, and $\tan^{-1}x$, evaluating them exactly at special-angle values and with a calculator, using them to find angles in right triangles, and evaluating compositions of trigonometric and inverse trigonometric functions.

The sections above are adapted from Precalculus 2e, Chapter 6 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org.