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 .
Answers
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.
- Graph the function from and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from .
Use the equation to answer the following questions.
- Graph the function from and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from .
Use the equation to answer the following questions.
- Graph the function from and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from .
- 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.
- Amplitude: -2 Period:
- Amplitude: Period:
- Amplitude: 9 Period: 6
- Challenge Find all the zeros from of .