Question

identify the trigonometric graph. question 2 2 pts

Explanation:

Step1: Analyze the properties of each trigonometric function

• \(\tan x\): The period of \(y = \tan x\) is \(\pi\), and its vertical asymptotes are at \(x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}\). But \(\tan x\) has a zero - crossing at \(x = n\pi,n\in\mathbb{Z}\).
• \(\sin x\): The function \(y=\sin x\) has a period of \(2\pi\), and \(y = \sin x=0\) when \(x = n\pi,n\in\mathbb{Z}\). Its range is \([- 1,1]\).
• \(\cos x\): The function \(y=\cos x\) has a period of \(2\pi\), and \(y = \cos x = 0\) when \(x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}\). Its range is \([-1,1]\).
• \(\sec x=\frac{1}{\cos x}\): The period of \(y = \sec x\) is \(2\pi\), and it has vertical asymptotes at \(x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}\).
• \(\csc x=\frac{1}{\sin x}\): The period of \(y=\csc x\) is \(2\pi\), and it has vertical asymptotes at \(x = n\pi,n\in\mathbb{Z}\).
• \(\cot x=\frac{\cos x}{\sin x}\): The period of \(y=\cot x\) is \(\pi\), and it has vertical asymptotes at \(x = n\pi,n\in\mathbb{Z}\), and \(y=\cot x = 0\) when \(x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}\).

Step2: Match the graph with the function

The given graph has a period of \(\pi\) (the distance between two consecutive vertical asymptotes or two consecutive \(x -\)intercepts is \(\pi\)) and \(x-\)intercepts at \(x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}\).

Answer:

\(\cot x\)