Tangent, Secant, Cosecant, and Cotangent

Tangent as a ratio, and what that costs

Define $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$. Everything unusual about the tangent graph follows from the denominator: wherever $\cos\theta=0$ — at $\theta=\frac{\pi}{2}+k\pi$ — the function is undefined and the graph has a vertical asymptote. Between consecutive asymptotes the graph is increasing and covers all of $\mathbb{R}$, so tangent has no amplitude and its period is $\pi$, not $2\pi$.

For $y=a\tan\big(b(x-c)\big)+d$: period $=\dfrac{\pi}{|b|}$, asymptotes solve $b(x-c)=\frac{\pi}{2}+k\pi$, and $d$ shifts the "midline point" where each branch crosses.

The reciprocal trio

• $\sec\theta=\frac{1}{\cos\theta}$ — asymptotes where $\cos\theta=0$; range $(-\infty,-1]\cup[1,\infty)$.
• $\csc\theta=\frac{1}{\sin\theta}$ — asymptotes where $\sin\theta=0$; same range.
• $\cot\theta=\frac{\cos\theta}{\sin\theta}$ — asymptotes where $\sin\theta=0$; decreasing on each branch, period $\pi$.

Sketching rule for sec and csc: draw the parent cosine or sine lightly, then flip each hump into a U-branch sitting on its max/min. The parent's zeros become the asymptotes; the parent's extrema become the vertices of the branches.