Trigonometry / Unit 4: Graphs of Trigonometric Functions
4.5 The Tangent Graph
Notes
The graph of $y = \tan x$ is very different from sine and cosine: it has vertical asymptotes and no maximum.
- Period $\pi$. Range: all real numbers. There is no amplitude.
- Vertical asymptotes at $\displaystyle x = \frac{\pi}{2} + n\pi$ (where $\cos x = 0$). Zeros at $x = n\pi$.
- The graph increases on each interval between asymptotes and passes through $\displaystyle \left(\frac{\pi}{4}, 1\right)$ and $\displaystyle \left(-\frac{\pi}{4}, -1\right)$.
- For $y = \tan(Bx)$ the period is $\displaystyle \frac{\pi}{|B|}$.
Practice
What is the period of $y = \tan x$?
Tangent repeats every $\pi$ units.
Which line is a vertical asymptote of $y = \tan x$?
$\displaystyle \cos\frac{\pi}{2} = 0$, so $\displaystyle \tan\frac{\pi}{2}$ is undefined.
What is the period of $y = \tan(2x)$?
Period $\displaystyle = \frac{\pi}{2}$ because $B = 2$.
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