Wize High School Algebra II Textbook (Common Core) > Trigonometric (Sinusoidal) Graphs

Graphs of Reciprocal Trigonometric Functions

Popular Courses

Algebra II

US High School

MA100

Wilfrid Laurier University

MATH 1149

Ohio State University

Find My Course

0:00 / 0:00

Graphs of Reciprocal Trigonometric Functions
The Cosecant Function
y=1sin⁡θ=csc⁡θy=\displaystyle\frac{1}{\sin\theta}=\csc{\theta}y=sinθ1​=cscθ

  

Maximums(3π2±2πn,−1),  n∈WMinimums(π2±2πn,1),  n∈WVertical Asymptotesθ=0±πn,  n∈WPeriod2πDomain{θ∈R∣ θ≠0±nπ,  n∈W}Rangey≤−1  &  y≥1\begin{array}{|c|c|}\hline\\
\textbf{Maximums}&\Bigg(\displaystyle\frac{3\pi}{2}\pm2\pi{}n,-1\Bigg),~~n\in\mathbb{W}\\\\\hline\\
\textbf{Minimums}&\Bigg(\displaystyle\frac{\pi}{2}\pm2\pi{}n,1\Bigg),~~n\in\mathbb{W}\\\\\hline\\
\textbf{Vertical Asymptotes}&\theta=0\pm{}\pi{}n,~~n\in\mathbb{W}\\\\\hline\\
\textbf{Period}&2\pi\\\\\hline\\
\textbf{Domain}&\{\theta\in\mathbb{R}|~\theta\neq{}0\pm{}n\pi,~~n\in\mathbb{W}\}\\\\\hline\\
\textbf{Range}&y\leq{}-1~~\&~~y\geq{}1\\\\\hline
\end{array}MaximumsMinimumsVertical AsymptotesPeriodDomainRange​(23π​±2πn,−1),  n∈W(2π​±2πn,1),  n∈Wθ=0±πn,  n∈W2π{θ∈R∣ θ=0±nπ,  n∈W}y≤−1  &  y≥1​​

PAGE BREAK
The Secant Function
y=1cos⁡θ=sec⁡θy=\displaystyle\frac{1}{\cos\theta}=\sec{\theta}y=cosθ1​=secθ
  

Maximums(π±2πn,−1),  n∈WMinimums(0±2πn,1),  n∈WVertical Asymptotesθ=π2±πn,  n∈WPeriod2πDomain{θ∈R∣ θ≠π2±nπ,  n∈W}Rangey≤−1  &  y≥1\begin{array}{|c|c|}\hline\\
\textbf{Maximums}&(\pi\pm2\pi{}n,-1),~~n\in\mathbb{W}\\\\\hline\\
\textbf{Minimums}&(0\pm2\pi{}n,1),~~n\in\mathbb{W}\\\\\hline\\
\textbf{Vertical Asymptotes}&\theta=\displaystyle\frac{\pi}{2}\pm{}\pi{}n,~~n\in\mathbb{W}\\\\\hline\\
\textbf{Period}&2\pi\\\\\hline\\
\textbf{Domain}&\{\theta\in\mathbb{R}|~\theta\neq{}\displaystyle\frac{\pi}{2}\pm{}n\pi,~~n\in\mathbb{W}\}\\\\\hline\\
\textbf{Range}&y\leq{}-1~~\&~~y\geq{}1\\\\\hline
\end{array}MaximumsMinimumsVertical AsymptotesPeriodDomainRange​(π±2πn,−1),  n∈W(0±2πn,1),  n∈Wθ=2π​±πn,  n∈W2π{θ∈R∣ θ=2π​±nπ,  n∈W}y≤−1  &  y≥1​​


Wize Tip
Maximums  on cos⁡θ, sin⁡θ\colorThree{\cos{\theta},~\sin{\theta}}cosθ, sinθ turn into minimums on sec⁡θ, csc⁡θ\colorThree{\sec{\theta},~\csc{\theta}}secθ, cscθ.
Minimums  on cos⁡θ, sin⁡θ\colorThree{\cos{\theta},~\sin{\theta}}cosθ, sinθ turn into maximums on sec⁡θ, csc⁡θ\colorThree{\sec{\theta},~\csc{\theta}}secθ, cscθ.
X-intercepts  on cos⁡θ, sin⁡θ\colorThree{\cos{\theta},~\sin{\theta}}cosθ, sinθ turn into vertical asymptotes on sec⁡θ, csc⁡θ\colorThree{\sec{\theta},~\csc{\theta}}secθ, cscθ.

PAGE BREAK
The Cotangent Function
y=1tan⁡θ=cot⁡θy=\displaystyle\frac{1}{\tan\theta}=\cot{\theta}y=tanθ1​=cotθ
  

Vertical Asymptotesθ=0±πn,  n∈WX-intercepts(π2±πn,0),  n∈WPeriodπDomain{θ∈R∣ θ≠0±nπ,  n∈W}Range−∞<y<∞\begin{array}{|c|c|}\hline\\
\textbf{Vertical Asymptotes}&\theta=0\pm{}\pi{}n,~~n\in\mathbb{W}\\\\\hline\\
\textbf{X-intercepts}&(\displaystyle\frac{\pi}{2}\pm{}\pi{}n,0),~~n\in\mathbb{W}\\\\\hline\\
\textbf{Period}&\pi\\\\\hline\\
\textbf{Domain}&\{\theta\in\mathbb{R}|~\theta\neq{}0\pm{}n\pi,~~n\in\mathbb{W}\}\\\\\hline\\
\textbf{Range}&-\infin<y<\infin\\\\\hline
\end{array}Vertical AsymptotesX-interceptsPeriodDomainRange​θ=0±πn,  n∈W(2π​±πn,0),  n∈Wπ{θ∈R∣ θ=0±nπ,  n∈W}−∞<y<∞​​

Wize Tip
X-intercepts  on tan⁡θ\colorThree{\tan{\theta}}tanθ turn into vertical asymptotes on cot⁡θ\colorThree{\cot{\theta}}cotθ.
Vertical asymptotes  on tan⁡θ\colorThree{\tan{\theta}}tanθ turn into x-intercepts on cot⁡θ\colorThree{\cot{\theta}}cotθ.

0:00 / 0:00

Example: Graphs of Reciprocal Trigonometric Functions
Graph y=2csc⁡(12(θ−π4))+1y=2\csc{\Bigg(\displaystyle\frac{1}{2}\Big(\theta-\displaystyle\frac{\pi}{4}\Big)\Bigg)+1}y=2csc(21​(θ−4π​))+1. 

First, graph the function y=2sin⁡(12(θ−π4))+1y=2\sin{\Bigg(\displaystyle\frac{1}{2}\Big(\theta-\displaystyle\frac{\pi}{4}\Big)\Bigg)+1}y=2sin(21​(θ−4π​))+1.
  
Turn all x-intercepts into vertical asymptotes. 
  
Turn all maximums into minimums. 
Turn all minimums into maximums. 

Erase  y=2sin⁡(12(θ−π4))+1\color{red}y=2\sin{\Bigg(\displaystyle\frac{1}{2}\Big(\theta-\displaystyle\frac{\pi}{4}\Big)\Bigg)+1}y=2sin(21​(θ−4π​))+1

0:00 / 0:00

Example: Graphs of Reciprocal Trigonometric Functions

Graph y=2cot⁡(2(θ−π16))y=2\cot{\Bigg(2\Big(\theta-\displaystyle\frac{\pi}{16}\Big)\Bigg)}y=2cot(2(θ−16π​)). 

First, graph y=2tan⁡(2(θ−π16))y=2\tan{\Bigg(2\Big(\theta-\displaystyle\frac{\pi}{16}\Big)\Bigg)}y=2tan(2(θ−16π​)).

Turn x-intercepts into vertical asymptotes.
Turn vertical asymptotes in x-intercepts.

Practice: Graphs of Reciprocal Trigonometric Functions

Find the vertical asymptotes of y=3sec⁡(3(θ−π18))+1y=3\sec\Bigg(3\Big(\theta-\displaystyle\frac{\pi}{18}\Big)\Bigg)+1y=3sec(3(θ−18π​))+1.

A) (2π9,0)±2πn, n∈W\Bigg(\displaystyle\frac{2\pi}{9},0\Bigg)\pm{}{2\pi}{n},~n\in\mathbb{W}(92π​,0)±2πn, n∈W 

B) (2π9,0)±π3n, n∈W\Bigg(\displaystyle\frac{2\pi}{9},0\Bigg)\pm{}\frac{\pi}{3}{n},~n\in\mathbb{W}(92π​,0)±3π​n, n∈W 

C)  (π6,0)±2π3n, n∈W\Bigg(\displaystyle\frac{\pi}{6},0\Bigg)\pm{}\frac{2\pi}{3}{n},~n\in\mathbb{W}(6π​,0)±32π​n, n∈W 

D) (π6,0)±2πn, n∈W\Bigg(\displaystyle\frac{\pi}{6},0\Bigg)\pm{}{2\pi}{n},~n\in\mathbb{W}(6π​,0)±2πn, n∈W

I don't know

Sketch a graph of the function y=2csc⁡(θ2−π2)y=2\csc\Bigg(\displaystyle\frac{\theta}{2}-\displaystyle\frac{\pi}{2}\Bigg)y=2csc(2θ​−2π​). 

Practice: Graphs of Reciprocal Trigonometric Functions
If the point (1,2)(1,2)(1,2) is on the graph of y=3cos⁡2θ+1y=3\cos{2\theta}+1y=3cos2θ+1, then what point must be on the graph y=sec⁡θy=\sec{\theta}y=secθ?

Answer

I don't know

Extra Practice

Graphs of Reciprocal Trigonometric Functions

Practice: Graphs of Reciprocal Trigonometric Functions
If the point (1,2)(1,2)(1,2) is on the graph of y=3cos⁡2θ+1y=3\cos{2\theta}+1y=3cos2θ+1, then what point must be on the graph y=sec⁡θy=\sec{\theta}y=secθ?

Graphs of Reciprocal Trigonometric Functions

Sketch a graph of the function y=2csc⁡(θ2−π2)y=2\csc\Bigg(\displaystyle\frac{\theta}{2}-\displaystyle\frac{\pi}{2}\Bigg)y=2csc(2θ​−2π​). 

Graphs of Reciprocal Trigonometric Functions

Practice: Graphs of Reciprocal Trigonometric Functions
Find the vertical asymptotes of y=3sec⁡(3(θ−π18))+1y=3\sec\Bigg(3\Big(\theta-\displaystyle\frac{\pi}{18}\Big)\Bigg)+1y=3sec(3(θ−18π​))+1.

Graphs of Reciprocal Trigonometric Functions

If f(x)=2cot⁡6(x−π24)+1f(x)=2\cot{6\Big(x-\dfrac{\pi}{24}\Big)}+1f(x)=2cot6(x−24π​)+1, where are the asymptotes?