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 their graphs which have direct application.
Subsection7.4.1Beyond Triangles
In triangles every angle had to be less than \(180^\circ\) because the sum of the angles of a triangle are only \(180^\circ\text{.}\) However, in many applications rather than measuring angles on objects we are measuring how far or how many times around something has moved. Use Activity 10 to explore this idea.
Activity10.
Use the the illustration in Figure 7.4.1 to see how angles are measured and how the trig functions act on larger angles. Find each of the following.
(a)
Angles of measure \(30^\circ\) and \(210^\circ\)
(i)
What is the sine value for both points?
(ii)
Compare the x coordinates of these two points.
(iii)
Where is the triangle created by the angle \(210^\circ\text{?}\)
(b)
Angles of measure \(45^\circ\) and \(315^\circ\)
(i)
What is the sine value for both points?
(ii)
Compare the sine values for these two points.
(iii)
Where is the triangle created by the angle \(315^\circ\text{?}\)
(c)
Angles of measure \(45^\circ\) and \(405^\circ\)
(i)
Compare the sine values for these two points.
(ii)
Where is the triangle created by the angle \(315^\circ\text{?}\)
(iii)
Note the angle displayed at the origin for \(405^\circ\text{:}\) why does it not match the slider angle?
(d)
Angles of measure \(-45^\circ\) and \(315^\circ\)
(i)
Where are the triangles for these two points?
(ii)
Move the slider from \(0^\circ\) to \(-45^\circ\text{.}\) Which direction does the point move?
(iii)
Note the angle displayed at the origin: explain why it is reasonable.
Use the slider to change the angle. Observe how the angle is drawn and how the sine and cosine values change.
You can drag the slider or click on it and use the arrow keys.
Figure7.4.1.Bigger Angles
We can use the definition of sine as a ratio and this understanding of angles to produce a graph. In Figure 7.4.2 drag the slider until you have the full graph. A graph that extends over a longer range (and labeled in degrees) is in Figure 7.4.3.
Use the slider to change the angle and produce the graph of sine. Observe how points on the graph corresponde to heights on the triangle.
Figure7.4.2.Produce Graph of Sine
Subsection7.4.2Properties of Sine Waves
Figure7.4.3.Graph of Sine
Notice that the graph of sine is a wave that repeats. The piece that repeats is called a cycle. In the default graph this is from \(0^\circ\) to \(360^\circ\) as shown in Figure 7.4.2.
The length of the cycle can be modified. Depending on the application we interpret and measure the length of the cycle differently.
Definition7.4.4.Period.
The length of a cycle measured in time is called the period.
Definition7.4.5.Wavelength.
The length of a cycle measured in distance is called the wavelength.
Sometimes instead of measuring how long a single cycle is units of time, we measure how many cycles occur in a fixed unit of time. This is called frequency.
Definition7.4.6.Frequency.
The number of waves (periods) that occur per second is called the frequency. This is typically measured in Hertz (Hz). 1 Hz is one cycle per second.
Note that frequency is the inverse of the period as shown in Table 7.4.7
Table7.4.7.Period and Frequency are Inverses
Period
Frequency
\(\frac{1\text{ cycle}}{n\text{ seconds}}\)
\(\frac{n\text{ cycles}}{1\text{ second}}\)
Example7.4.8.
If a wave has a period of \(1/3\) seconds, what is its frequency?
Because we know the frequency we can directly calculate the period.
\begin{align*}
261.63 \amp = \frac{1}{T}\\
T \cdot 261.63 \amp = T \cdot \frac{1}{T}\\
\frac{T \cdot 261.63}{261.33} \amp = \frac{1}{261.33}\\
T \amp = \frac{1}{261.63}\\
T \amp \approx 0.0038222
\end{align*}
For the wavelength we need to recall 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).
A local AM radio station broadcasts at \(750.0\) Hz. Note radio wave 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?
Because we know the frequency we can directly calculate the period.
\begin{equation*}
\begin{aligned}
293.66 \amp = \frac{1}{T}\\
T \cdot 293.66 \amp = T \cdot \frac{1}{T}\\
\frac{T \cdot 293.66}{293.66} \amp = \frac{1}{293.66}\\
T \amp = \frac{1}{293.66}\\
T \amp \approx 0.0034053
\end{aligned}
\end{equation*}
For the wavelength we need to recall 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).
In Subsection 5.4.2 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. Here we will learn to change the amplitude and the frequency of sine waves.
