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Tangent Graphs

Adjust the length of the curve, or the distance before the y values repeat, from 2pi.

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Changes in the Period of a Sine and Cosine Function

What is the period of the cosine function ?

Guidance

The last thing that we can manipulate on the sine and cosine curve is the period.

The normal period of a sine or cosine curve is . To stretch out the curve, then the period would have to be longer than . Below we have sine curves with a period of and then the second has a period of .

To determine the period from an equation, we introduce into the general equation. So, the equations are and , where is the amplitude, is the frequency, is the phase shift, and is the vertical shift. The frequency is the number of times the sine or cosine curve repeats within . Therefore, the frequency and the period are indirectly related. For the first sine curve, there is half of a sine curve in . Therefore the equation would be . The second sine curve has two curves within , making the equation . To find the period of any sine or cosine function, use , where is the frequency. Using the first graph above, this is a valid formula: .

Example A

Determine the period of the following sine and cosine functions.

a)

b)

c)

Solution: a) The 6 in the equation tells us that there are 6 repetitions within . So, the period is .

b) The in the equation tells us the frequency. The period is .

c) The is the frequency. The period is .

Example B

Graph part a) from the previous example from . Determine where the maximum and minimum values occur. Then, state the domain and range.

Solution: The amplitude is -3, so it will be stretched and flipped. The period is (from above) and the curve should repeat itself 6 times from 0 to . The first maximum value is 3 and occurs at half the period, or and then repeats at Writing this as a formula we start at and add to get the next maximum, so each point would be where is any integer.

The minimums occur at -3 and the -values are multiples of . The points would be , again is any integer. The domain is all real numbers and the range is .

Example C

Find all the solutions from the function in Example B from .

Solution: Before this concept, the zeros didn’t change in the frequency because we hadn’t changed the period. Now that the period can be different, we can have a different number of zeros within . In this case, we will have 6 times the number of zeros that the parent function. To solve this function, set and solve for .

Now, use the inverse cosine function to determine when the cosine is zero. This occurs at the multiples of .

We went much past because when we divide by 6, to get by itself, all of these answers are going to also be divided by 6 and smaller.

so we have found all the zeros in the range.

Concept Problem Revisit First, we need to get the function in the form . Therefore we need to factor out the 2.

The is the frequency. The period is therefore .

Guided Practice

1. Determine the period of the function .

2. Find the zeros of the function from #1 from .

3. Determine the equation of the sine function with an amplitude of -3 and a period of .

1. The period is .

2. The zeros would be when is zero.

3. The general equation of a sine curve is . We know that and that the period is . Let’s use this to find the frequency, or .

The equation of the curve is .

Vocabulary

Period
The length in which an entire sine or cosine curve is completed.
Frequency
The number of times a curve is repeated within .

Practice

Find the period of the following sine and cosine functions.

Use the equation to answer the following questions.

1. Graph the function from and find the domain and range.
2. Determine the coordinates of the maximum and minimum values.
3. Find all the zeros from .

Use the equation to answer the following questions.

1. Graph the function from and find the domain and range.
2. Determine the coordinates of the maximum and minimum values.
3. Find all the zeros from .

Use the equation to answer the following questions.

1. Graph the function from and find the domain and range.
2. Determine the coordinates of the maximum and minimum values.
3. Find all the zeros from .
4. What is the domain of every sine and cosine function? Can you make a general rule for the range? If so, state it.

Write the equation of the sine function, in the form , with the given amplitude and period.

1. Amplitude: -2 Period:
2. Amplitude: Period:
3. Amplitude: 9 Period: 6
4. Challenge Find all the zeros from of .