We began by looking at trigonometric functions in the context of triangles where they represent the ratio of side lengths. Here we will consider trigonometric functions in the context of properties of their graphs. The graphs have direct applications.
This section illustrates how trigonometric functions can be defined on angles greater than 180°. In triangles every angle had to be less than 180° because the sum of the angles of a triangle are only 180°. However, in many applications rather than measuring angles on objects (like triangles), we are measuring how far or how many times around something has moved. For example a wheel on a car moves more than 180° when we are driving. Use Activity 17 to explore this idea.
Use the the illustration in Figure 7.4.1 to see how angles, including those bigger than 180°, are measured and how the trig functions act on these angles.
We can use the definition of sine as a ratio and this understanding of angles to produce a graph. In Figure 7.4.3 drag the slider until you have the full graph. A graph that extends over a longer range (and labeled in degrees) is in Figure 7.4.4.
The horizontal axis ranges from -180 to 720 in increments of 90. The vertical axis ranges from -1 to 1. The graph of sine on this portion consists of 5 humps that are 180 wide and alternate down (starting at -180) and up. From 0 to 720 there are two complete up/down pairs.
From the first section we know that the graph of sine is a wave that repeats. The piece that is repeated is called a cycle. In the default graph this is from 0° to 360° as shown in Figure 7.4.3.
The length of the cycle can be modified. Depending on the application we interpret and measure the length of the cycle differently. This section defines two traits of the cycle, illustrates the traits, and gives an alternate definition for one trait.
The video below illustrates the difference between wavelength and period of a sine wave. The top two sine curves have the same wavelength (2), and the bottom two sine curves have the same wavelength (1). We can see this because the top two curves have one cycle from 0 to 2, and the bottom two curves have one cycle from 0 to 1. This implies the bottom curves have a wavelength that is half that of those on top.
The left two sine curves have the same period. They complete one cycle in one (1) seconds. The right two sine curves have the same period. They complete one cycle in two (2) seconds.
Sometimes instead of measuring how long a single cycle is in units of time, we measure how many cycles occur in a fixed unit of time. This is called frequency.
Note that frequency is the inverse of the period as shown below. In the video the sine waves above, the sine waves on the right have a period of two (2) and a frequency of \(1/2\text{.}\)
What are the period and wavelength of middle C which has a frequency of 261.63 Hz? This is a measurement in a science model so we will use significant digits.
For the wavelength we need to know that the speed of sound is 1116 feet/second. Now we can use the fact that Hz is cycles per second to convert frequency (cycles per second) to wavelength (feet per cycle). The units suggest that we multiply as follows.
A local AM radio station broadcasts at \(750.0\) Hz. Note radio waves move at the speed of light which is approximately \(2.9979 \times 10^8\) meters per second. What are the period and wavelength of this radio signal?
The bottom of the sine wave reaches -2 and the top reaches 2. A complete cycle is repeated from -180 to 0, again from 0 to 180, again from 180 to 360, again from 360 to 540, and again from 540 to 720.
In applications the wavelength, period/frequency, and amplitude are determined experimentally: we rarely see the wave. However, in order to practice distinguishing between these properties, we will use visual examples and exercises.
In Subsection 5.4.1 and Subsection 6.2.2 we learned how to transform a graph by shifting it and reflecting it. Those apply to trigonometric graphs as well. This section illustrates vertical and horizontal stretches and connects them to amplitude and wavelength.
This activity demonstrates how the sine function is modified to change amplitude. Use Figure 7.4.18 to answer the following. Note the amplitude of the unmodified graph is 1.
This activity demonstrates how the sine function is modified to change wavelength. Use Figure 7.4.18 to answer the following. Note the wavelength of the unmodified graph is 2.
