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Q29.

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Found in: Page 767

### Algebra 2

Book edition Middle English Edition
Author(s) Carter
Pages 804 pages
ISBN 9780079039903

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

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

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

See the step by step solution

## Step 1. Write down the given information.

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

## Step 2. Concept used.

A function of the form $y=a\mathrm{csc}\left(bx\right)\text{and}y=a\mathrm{sec}\left(bx\right)$ 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 $y=3\mathrm{csc}\left(\frac{\theta }{2}\right)$ is evaluated as:

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

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

## Step 4. Sketch the graph for the function.

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

## Step 5. Conclusion.

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

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