How to Graph the Cosecant Function?

A step-by-step guide to graphing the cosecant function

The cosecant function is the reciprocal of the trigonometric function \(sin\). Since the cosecant function is the reciprocal of the \(sin\) function, we can write its formula as:

\(\color{blue}{csc (\theta)=\frac{Hypotenuse}{opposite\: side}=\frac{1}{sin\:\theta}}\)

\(cosec x\) is defined for all real numbers except for values where \(sin x\) is equal to zero. Therefore, we have vertical asymptotes at points where \(csc x\) is not defined. Also, using the values of \(sin x\), we have \(y=csc x\) as:

• When \(x = 0\), \(sin x = 0\) \(\rightarrow\) \(csc x =\) not defined
• When \(x = \frac{\pi }{6}\), \(sin x = \frac{1}{2}\)\(\rightarrow\) \(csc x = 2\)
• When \(x =\frac{\pi }{4}\), \(sin x=\frac{1}{\sqrt{2}}\)\(\rightarrow\) \(csc x = \sqrt{2}\)
• When \(x =\frac{\pi }{3}\), \(sin x= \frac{\sqrt{3}}{2}\)\(\rightarrow\) \(csc x = \frac{2}{\sqrt{3}}\)
• When \(x =\frac{\pi }{2}\), \(sin x = 1\)\(\rightarrow\) \(csc x=1\)

Therefore, by drawing the above points on a graph and connecting them, we have the cosecant graph as follows:

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