Q30. - Algebra 2

Question

Find the amplitude if it exists and period of the function $y=\frac{1}{2}\cot \left(2\theta \right)$ and then graph the function.

Step-by-Step Solution

Answer

The amplitude of $y=\frac{1}{2}\cot \left(2\theta \right)$ is undefined.

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

Step 1. Write down the given information.

The given function is $y=\frac{1}{2}\cot \left(2\theta \right)$.

Step 2. Concept used.

A function of the form $y=a\tan \left(bx\right) \text{and }y=a\cot \left(bx\right)$ has undefined amplitude and period $\frac{180°}{\left|b\right|} \text{or }\frac{\pi }{\left|b\right|}$.

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

With the help of concept stated above, the amplitude and period of the function $y=\frac{1}{2}\cot \left(2\theta \right)$ is evaluated as:

The amplitude of $y=\frac{1}{2}\cot \left(2\theta \right)$ is undefined.

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

Step 4. Sketch the graph for the function.

The graph for the function $y=\frac{1}{2}\cot \left(2\theta \right)$ is shown below.

Step 5. Conclusion.

The amplitude of $y=\frac{1}{2}\cot \left(2\theta \right)$ is undefined.

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