Precalculus / Unit 4: Trigonometric Functions
4.5 Other Trigonometric Graphs
Notes
- $\tan x$ has period $\pi$ and vertical asymptotes at $\displaystyle x = \frac{\pi}{2} + k\pi$. For $y = \tan(Bx)$ the period is $\displaystyle \frac{\pi}{|B|}$.
- $\displaystyle \csc x = \frac{1}{\sin x}$ and $\displaystyle \sec x = \frac{1}{\cos x}$ have asymptotes wherever the denominator is zero.
- The range of $\sec x$ and $\csc x$ is $(-\infty, -1] \cup [1, \infty)$.
- $\displaystyle \cot x = \frac{\cos x}{\sin x}$ has period $\pi$ and asymptotes at $x = k\pi$.
Practice
What is the period of $y = \tan(2x)$?
Period $\displaystyle = \frac{\pi}{2}$ since $B = 2$.
Which function has a vertical asymptote at $\displaystyle x = \frac{\pi}{2}$?
$\displaystyle \tan x = \frac{\sin x}{\cos x}$ and $\displaystyle \cos\frac{\pi}{2} = 0$.
What is the range of $y = \sec x$?
$\displaystyle \sec x = \frac{1}{\cos x}$ and $|\cos x| \le 1$, so $|\sec x| \ge 1$.
Ready for more? Take the Unit 4 test.