## Precalculus (6th Edition) Blitzer We know that the cosecant function is the reciprocal of the sine function, so graph the reciprocal sine function $y=\frac{5}{2}\sin \left( x-\pi \right)$. The given equation is in the form of $y=A\sin \left( Bx-C \right)$. Here, $A=-2,\text{ }B=\pi \text{ and }C=\pi$ So, the amplitude is: \begin{align} & \text{Amplitude=}\left| A \right| \\ & =\left| \frac{5}{2} \right| \\ & =\frac{5}{2} \end{align} The period is given below: \begin{align} & \text{Period = }\frac{2\pi }{B} \\ & =\frac{2\pi }{1} \\ & =2\pi \end{align} And the phase shift is: \begin{align} & \text{Phase shift=}\frac{C}{B} \\ & =\frac{\pi }{1} \\ & =\pi \end{align} And the quarter period is as follows: \begin{align} & \text{Quarter-period}=\frac{2\pi }{4} \\ & =\frac{\pi }{2} \end{align} Now, add quarter periods starting from $x=\pi$ to generate x-values for the key points. The x-value for the first key point is as follows: $x=\pi$ And the x-value for the second key point is: \begin{align} & x=\pi +\frac{\pi }{2} \\ & =\frac{3\pi }{2} \end{align} And the x-value for the third key point is: \begin{align} & x=\frac{3\pi }{2}+\frac{\pi }{2} \\ & =2\pi \end{align} And the x-value for the fourth key point is: \begin{align} & x=2\pi +\frac{\pi }{2} \\ & =\frac{5\pi }{2} \end{align} And the x-value for the fifth key point is: \begin{align} & x=\frac{5\pi }{2}+\frac{\pi }{2} \\ & =3\pi \end{align} Now, draw the graph by connecting the five key points with a smooth dotted curve for one complete cycle and then extend the graph one cycle to the right. Now, use this graph to draw the graph of the reciprocal function by drawing vertical asymptotes through the x-intercepts, and use them as guides to graph $y=\frac{5}{2}\csc \left( x-\pi \right)$.