Question

Find the amplitude if it exists and period of the function $y=3\csc \left(\frac{\theta }{2}\right)$ and then graph the function.

Short answer

The amplitude of $y=3\csc \left(\frac{\theta }{2}\right)$is undefined.

The period of $y=3\csc \left(\frac{\theta }{2}\right)$ is $4\pi$.

Step-by-step solution

Step 1. Write down the given information.

The given function is $y=3\csc \left(\frac{\theta }{2}\right)$.

Step 2. Concept used.

A function of the form $y=a\csc \left(bx\right)\text{and}y=a\sec \left(bx\right)$ has undefined amplitude and period $\frac{360°}{\left|b\right|}\text{or}\frac{2\pi }{\left|b\right|}$.

Step 3. Evaluating amplitude and period of the given function.

With the help of concept stated above, the amplitude and period of the function $y=3\csc \left(\frac{\theta }{2}\right)$ is evaluated as:

The amplitude of $y=3\csc \left(\frac{\theta }{2}\right)$is undefined.

The period of $y=3\csc \left(\frac{\theta }{2}\right)$is $\frac{2\pi }{\left|\frac{1}{2}\right|}=4\pi$.

Step 4. Sketch the graph for the function.

The graph for the function $y=3\csc \left(\frac{\theta }{2}\right)$ is shown below.

Step 5. Conclusion.

The amplitude of $y=3\csc \left(\frac{\theta }{2}\right)$is undefined.

The period of $y=3\csc \left(\frac{\theta }{2}\right)$ is $4\pi$.