# High School Math : Trigonometric Functions and Graphs

## Example Questions

### Example Question #1 : Trigonometric Functions And Graphs

Which of the following is not in the range of the function ?

Explanation:

The range of the function  is all numbers between  and  (the sine wave never goes above or below this).

Of the choices given,  is greater than  and thus not in this range.

### Example Question #2 : Trigonometric Functions And Graphs

Simplify the function below:

Explanation:

We need to use the following formulas:

a)

and

b)

We can simplify as follows:

### Example Question #3 : Trigonometric Functions And Graphs

What is the period of

?

Explanation:

The period for  is . However, if a number is multiplied by , you divide the period  by what is being multiplied by . Here,  is being multiplied by    equals

### Example Question #4 : Trigonometric Functions And Graphs

What is the amplitude of ?

Explanation:

The amplitude of a wave function like  is always going to be the coefficient of the function. In this case, that is .

### Example Question #5 : Trigonometric Functions And Graphs

What is the local maximum of  between  and ?

Explanation:

The fastest way to solve this problem is to graph it and observe the answer. However, the other option is to think of this equation in terms of period.

When the coefficient of the variable increases, the frequency increases and the period decreases by that rate.

Since our equation is , our period will be  the normal period of a  wave. Since only the period is changing, the amplitude is not. Therefore the amplitude (the highest and lowest points) of  will be the same as that of . The amplitude of a sine wave is , so the amplitude of  will also be .

Therefore, our maximum will be .

### Example Question #2 : Period And Amplitude

Which of the given functions has the greatest amplitude?

Explanation:

The amplitude of a function is the amount by which the graph of the function travels above and below its midline. When graphing a sine function, the value of the amplitude is equivalent to the value of the coefficient of the sine. Similarly, the coefficient associated with the x-value is related to the function's period. The largest coefficient associated with the sine in the provided functions is 2; therefore the correct answer is .

The amplitude is dictated by the coefficient of the trigonometric function. In this case, all of the other functions have a coefficient of one or one-half.