2.15: Period and Frequency
While working on an assignment about sound in your science class, your Instructor informs you that what you know as the "pitch" of a sound is, in fact, the frequency of the sound waves. He then plays a note on a musical instrument, and the pattern of the sound wave on a graph looks like this:
He then tells you to find the frequency of the sound wave from the graph? Can you do it?
Don't worry. By the end of this Concept, you'll understand what frequency is and be able to find it from a plot like this one.
Watch This
In the second part of this video you'll learn about the period of trigonometric functions.
James Sousa: Amplitude and Period of Sine and Cosine
Guidance
The period of a trigonometric function is the horizontal distance traversed before the
Frequency is a measurement that is closely related to period. In science, the frequency of a sound or light wave is the number of complete waves for a given time period (like seconds). In trigonometry, because all of these periodic functions are based on the unit circle, we usually measure frequency as the number of complete waves every
Period and frequency are inversely related. That is, the higher the frequency (more waves over
After observing the transformations that result from multiplying a number in front of the sinusoid, it seems natural to look at what happens if we multiply a constant inside the argument of the function, or in other words, by the
Notice that the number of waves for
Example A
What is the frequency and period of
Solution: If we follow the pattern from the previous example, multiplying the angle by 3 should result in the sine wave completing a cycle three times as often as
This number that is multiplied by
Adding, one last time to our equations from before, we now have:
Example B
What is the frequency and period of
Solution: Using the generalization above, the frequency must be
Thinking of it as a transformation, the graph is stretched horizontally. We would only see
Example C
What is the frequency and period of
Solution:
Like the previous two examples, we can see that the frequency is
Vocabulary
Period: The period of a wave is the horizontal distance traveled before the 'y' values begin to repeat.
Frequency: The frequency of a wave is number of complete waves every
Guided Practice
1. Draw a sketch of
2. Draw a sketch of
3. Draw a sketch of
Solutions:
1. The "2" inside the sine function makes the function "squashed" by a factor of 2 in the horizontal direction.
2. The
3. The
Concept Problem Solution
By inspecting the graph
You can see that the wave takes about 6.2 seconds to make one complete cycle. This means that the frequency of the wave is approximately 1 cycle per second (since
Practice
Find the period and frequency of each function below.

y=sin(4x) 
y=cos(2x) 
y=cos(12x) 
y=sin(34x) 
y=sin(3x)
Draw a sketch of each function from 0 to

y=sin(3x) 
y=cos(5x) 
y=3cos(25x) 
y=12sin(34x) 
y=−sin(2x) 
y=tan(3x) 
y=sec(2x) 
y=csc(4x)
Find the equation of each function.
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Term  Definition 

Amplitude  The amplitude of a wave is onehalf of the difference between the minimum and maximum values of the wave, it can be related to the radius of a circle. 
Frequency  The frequency of a trigonometric function is the number of cycles it completes every units. 
horizontal stretch  Horizontal stretch describes the stretching of a graph from the axis. For a sinusoidal function with equation , the coefficient controls horizontal stretch. 
Period  The period of a wave is the horizontal distance traveled before the values begin to repeat. 
sinusoidal function  A sinusoidal function is a sine or cosine wave. 
sinusoidal functions  A sinusoidal function is a sine or cosine wave. 
Vertical shift  A vertical shift is the result of adding a constant term to the value of a function. A positive term results in an upward shift, and a negative term in a downward shift. 
Image Attributions
Here you'll learn how to find the period and frequency of a trig function from either the graph or the algebraic equation.