Graphs of Trigonometric Functions

Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes where cosine is zero. Conversely, the cotangent function is the ratio of cosine to sine and behaves similarly but with asymptotes where sine is zero.

The secant and cosecant functions, which are the reciprocals of cosine and sine, respectively, form curves that arch away from the x-axis, becoming undefined at the zeros of their base functions. These graphs display vertical asymptotes and do not intersect the x-axis. Despite their different appearances, all these functions exhibit regular periodicity and distinct symmetries—sine, tangent, and cosecant being odd functions symmetric about the origin, while cosine and secant are even functions symmetric about the y-axis. Their domain restrictions arise from division by zero in their definitions, and their ranges reflect whether they are bounded or unbounded.

Graph Properties of Trigonometric Functions