The same reflection rule applies to asin bx+c, acos bx+c, and atan bx+c, with positive integer a, integer c, and simple fractional or integer b. In Cambridge fractional frequencies have denominator 2,3,4,6 or 8. For |2sin x-1|, 0 <= x 2pi, zeros occur at pi/6 and 5pi/6. Reflect the negative sections; the range becomes [0,3]. For |cos x|, the zeros are pi/2,3pi/2 and the period is pi. For |tan x|, retain the vertical asymptotes x=pi/2+kpi; reflecting outputs never fills a missing input. Use separate branches near each asymptote.

The same reflection rule applies to \(a\sin bx+c\), \(a\cos bx+c\), and \(a\tan bx+c\), with positive integer \(a\), integer \(c\), and simple fractional or integer \(b\). In Cambridge fractional frequencies have denominator \(2,3,4,6\) or \(8\).

For \(|2\sin x-1|\), \(0\le x\le2\pi\), zeros occur at \(\pi/6\) and \(5\pi/6\). Reflect the negative sections; the range becomes \([0,3]\). For \(|\cos x|\), the zeros are \(\pi/2,3\pi/2\) and the period is \(\pi\).

For \(|\tan x|\), retain the vertical asymptotes \(x=\pi/2+k\pi\); reflecting outputs never fills a missing input. Use separate branches near each asymptote.

Moduli of trigonometric graphs

Moduli of trigonometric graphs