2.2 · Trigonometric graphs, inverses, and periodic models

Separate amplitude, midline, and period

For $\displaystyle y=A\sin(B(x-C))+D$, the amplitude is $\displaystyle |A|$, midline is $\displaystyle y=D$, period is $\displaystyle 2\pi/|B|$, and phase shift is $\displaystyle C$. Factor the input before reading the shift.

Maximum and minimum heights determine the midline as their average and amplitude as half their difference. The horizontal distance between consecutive peaks gives the period.

$$T=\frac{2\pi}{|B|}$$

Read tangent and reciprocal-function restrictions

Tangent has period $\displaystyle \pi$ and vertical asymptotes where cosine is zero. Secant and cosecant have branches separated by zeros of cosine or sine. Their ranges exclude values strictly between $\displaystyle -1$ and $\displaystyle 1$ before transformations.

A graphing window can hide asymptotes or suggest false connecting segments. Use the defining denominator zeros to confirm the domain and branch breaks.

An inverse function needs a restricted range

The principal range of $\displaystyle \arcsin$ is $\displaystyle [-\pi/2,\pi/2]$, of $\displaystyle \arccos$ is $\displaystyle [0,\pi]$, and of $\displaystyle \arctan$ is $\displaystyle (-\pi/2,\pi/2)$. These restrictions make the inverses single-valued.

For $\displaystyle \arcsin(\sin(5\pi/6))$, the inner sine is $\displaystyle 1/2$ and the principal inverse output is $\displaystyle \pi/6$, not $\displaystyle 5\pi/6$. An inverse symbol does not cancel indiscriminately outside the restricted interval.