Question

How do you graph $y=\csc \left( 2x-\dfrac{\pi }{2} \right)$?

Answer 1

In this question we have the trigonometric expression in terms of cosecant. We will use the general equation of a secant function. the general cosecant function is in the form of $y=a\csc (bx+c)+d$, where $a$ is the amplitude of the function, $b$ is to found the period of the function which has the formula $\dfrac{2\pi }{\left| b \right|}$, $c$ represents the shift of the function and $d$ represents the baseline of the function.

Complete step by step answer:
We have the given function as:
$\Rightarrow y=\csc \left( 2x-\dfrac{\pi }{2} \right)$
Now on comparing the given function with the general form of a secant function which is $y=a\csc (bx+c)+d$, we get:
$a=1$, which means there is no change in the amplitude.
The period of the graph can be calculated as $\dfrac{2\pi }{\left| b \right|}$.
On substituting, we get $\dfrac{2\pi }{2}$.
On simplifying, we get the period as $\pi$.
On using the scientific calculator to calculate the value of $\pi$, we get $3.142$ which is the period.
The shift of the function $c=-\dfrac{\pi }{2}$ and the baseline of the function $d=0$.
Therefore, given all the values the graph can be plotted as:

Which is the required cosecant function $y=\csc \left( 2x-\dfrac{\pi }{2} \right)$.

Note: In this question we are using the cosecant function. There also exists the sine function and cosine function which is represented as: $y=a\sin (bx+c)+d$ and $y=a\cos (bx+c)+d$, which have the same properties that of a tangent function.
The sign of the shift $c$ represents in which direction the shift is taking place, it could be negative or positive for right and left respectively.