Chapter 13.7, Problem 5CYU

a.

To determine

To determine the period of the given function.

Answer to Problem 5CYU

  $\frac{1}{14}$

Explanation of Solution

Given:

Frequency $=14$ hertz.

Concept used:

The generalized equation for a trigonometric function is:

  $y=Af\left(Bx+C\right)+D$

Period $=\frac{1}{\text{Frequency}}$

Calculation:

The period of the given function $=\frac{1}{\text{Frequency}}$ $=\frac{1}{14}$

Conclusion:

Therefore, the period of the given function is $\frac{1}{14}$ .

b.

To determine

To determine the sine equation and graph it.

Answer to Problem 5CYU

  $y=\sin 28\pi \theta$

Explanation of Solution

Given:

Amplitude $=1$ unit

Frequency $=14$ hertz.

Concept used:

The generalized equation for a cosine function is:

  $y=A\sin Bx$

Amplitude $=A$

Period $=\frac{2\pi }{B}$

  $\Rightarrow$ $B=\frac{2\pi }{\text{Period}}$

Period $=\frac{1}{\text{Frequency}}$

Calculation:

Amplitude tells that the graph will be one time taller than the usual.

  $A=1$

The period of the given function $=\frac{1}{\text{Frequency}}$

  $=\frac{1}{14}$

So, $B=\frac{2\pi }{\text{Period}}$

  $B=\frac{2\pi }{\frac{1}{14}}$

  $\begin{array}{l}=2\pi \times 14\\ =28\pi \end{array}$

Therefore, the cosine equation is $y=\sin 28\pi \theta$ .

Graph:

The graph of $y=\sin 28\pi \theta :$

  

Conclusion:

Therefore, the sine function is $y=\sin 28\pi \theta$ .