# Trigonometry : Period and Amplitude

## Example Questions

### Example Question #1 : Period And Amplitude

What is the amplitude in the graph of the following equation:

Explanation:

The general form for a sine equation is:

The amplitude of a sine equation is the absolute value of .

Since our equation begins with , we would simplify the equation:

The absolute value of  would be .

### Example Question #1 : Period And Amplitude

What is the amplitude of

Explanation:

Amplitude describes the distance from the middle of a periodic function to its local maximum.  covers the range from -1 to 1. Thus, it covers a distance of 2 vertically. Half of this, or 1, gives us the amplitude of the function. It is often helpful to think of the amplitude of a periodic function as its "height".

### Example Question #1 : Period And Amplitude

What is the amplitude of

Explanation:

The amplitude of a function describes its height from the midline to the maximum. The amplitude of the parent function, , is 1, since it goes from -1 to 1. In this case our function has been multiplied by 4. Think of the effects this multiplication has on the outputs. In , we get our maximum at , and . Here, we will get 4. The same thing happens for our minimum, at  , . Here, we get -4. Thus, by this analysis, it is clear that the amplitude is 4. In the future, remember that the number preceding the cosine function will always be its amplitude.

### Example Question #4 : Period And Amplitude

What is the period of the function

Explanation:

By definition, the period of a function is the length of  for which it repeats.  starts at 0, continues to 1, goes back to 0, goes to -1, and then back to 0.

This complete cycle goes from  to  .

### Example Question #1 : Period And Amplitude

What is the period and amplitude of the following trigonometric function?

Explanation:

Recall the form of a sinusoid:

or

The important quantities for this question are the amplitude, given by , and period given by  .

For this problem, amplitude is equal to  and period is .

### Example Question #1 : Period And Amplitude

What is the period of the following function?

Explanation:

The period of the standard cosine function is .

We can find the period of the given function by dividing  by the coefficient in front of , which is :

.

### Example Question #1 : Period And Amplitude

Write the equation of sine graph with amplitude 3 and period of

None of the above

Explanation:

Giving

,

where

and

Then,

,

hence

.

.

Therefore,

### Example Question #1 : Understanding Period And Amplitude

Which of the given functions has the greatest amplitude?