Graphs of Tangent, Cotangent, Secant, and Cosecant
Graph the four remaining trigonometric functions, identifying vertical asymptotes, period, and key features. Understand how each is derived from sine and cosine and how transformations apply.
A lesson of Algebra 2, in Introduction to Trigonometry, about 10 minutes.
By the end of this lesson, you will be able to sketch the graphs of all four remaining trig functions, identify their periods, vertical asymptotes, domains, and ranges, and apply transformations to each.
The Tangent Function
$$y = \tan x = \frac{\sin x}{\cos x}$$
Vertical asymptotes where $\cos x = 0$: at $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \ldots$ (i.e., $x = \frac{\pi}{2} + n\pi$)
Period: $\pi$ (half the period of sin/cos)
Domain: all $x \neq \frac{\pi}{2} + n\pi$
Range: $(-\infty, \infty)$
Key points per cycle: $(-\pi/4, -1)$, $(0, 0)$, $(\pi/4, 1)$
Behavior: increasing from $-\infty$ to $+\infty$ within each period.
The Cotangent Function
$$y = \cot x = \frac{\cos x}{\sin x}$$
Asymptotes where $\sin x = 0$: at $x = 0, \pm\pi, \pm 2\pi, \ldots$ (i.e., $x = n\pi$)
Period: $\pi$
Range: $(-\infty, \infty)$
Behavior: decreasing from $+\infty$ to $-\infty$ within each period. (Opposite direction from tangent.)
The Secant Function
$$y = \sec x = \frac{1}{\cos x}$$
Graph: built from the reciprocal of cosine. Where cosine = 0 → asymptotes; where cosine = ±1 → secant touches the cosine graph.
Asymptotes: same as tangent — $x = \frac{\pi}{2} + n\pi$
Period: $2\pi$
Range: $(-\infty, -1] \cup [1, \infty)$ — never between $-1$ and $1$
Shape: U-shapes opening up (from cosine maxima) and inverted U-shapes (from cosine minima).
The Cosecant Function
$$y = \csc x = \frac{1}{\sin x}$$
Asymptotes where $\sin x = 0$: $x = n\pi$
Period: $2\pi$
Range: $(-\infty, -1] \cup [1, \infty)$
Shape: Same U-shape pattern, but built from the reciprocal of sine.
Graphing Strategy: Use the Reciprocal
To graph sec or csc: