# 4.6 | Graphs of Other Trigonometric Functions

The graph $$y = a \tan (bx -c) + d$$ has the following characteristics.

• The amplitude is not defined
• The period is given by $$\displaystyle \frac{ \pi}{b}$$.
• Two consecutive vertical asymptotes can be found by solving the equations $$bx-c = -\frac{\pi}{2}$$ and $$bx-c = \frac{\pi}{2}$$
• The domain excludes all values of $$x$$ which are solutions to $$\cos(bx-c)=0.$$ That is, $$D = \lbrace x \in \mathbb{R} \; | \; \cos(bx-c) \not = 0 \rbrace$$

The graph $$y = a \cot (bx -c) + d$$ has the following characteristics.

• The amplitude is not defined
• The period is given by $$\displaystyle \frac{ \pi}{b}$$.
• Two consecutive vertical asymptotes can be found by solving the equations $$bx-c = 0$$ and $$bx-c = \pi$$
• The domain excludes all values of $$x$$ which are solutions to $$\sin(bx-c)=0.$$ That is, $$D = \lbrace x \in \mathbb{R} \; | \; \sin(bx-c) \not = 0 \rbrace$$

The graph $$y = a \sec (bx -c) + d$$ has the following characteristics.

• The amplitude is not defined
• The period is given by $$\displaystyle \frac{2 \pi}{b}$$.
• Two consecutive vertical asymptotes can be found by solving the equations $$bx-c = -\frac{\pi}{2}$$ and $$bx-c = \frac{\pi}{2}$$
• The domain excludes all values of $$x$$ which are solutions to $$\cos(bx-c)=0.$$ That is, $$D = \lbrace x \in \mathbb{R} \; | \; \cos(bx-c) \not = 0 \rbrace$$

The graph $$y = a \csc (bx -c) + d$$ has the following characteristics.

• The amplitude is not defined
• The period is given by $$\displaystyle \frac{2 \pi}{b}$$.
• Two consecutive vertical asymptotes can be found by solving the equations $$bx-c = 0$$ and $$bx-c = 2 \pi$$
• The domain excludes all values of $$x$$ which are solutions to $$\sin(bx-c)=0.$$ That is, $$D = \lbrace x \in \mathbb{R} \; | \; \sin(bx-c) \not = 0 \rbrace$$