Section 4.2 Graphs of Trigonometric Functions
Subsection A Periodic Function of Angle
Imagine that you are riding on a Ferris wheel. As the wheel turns, your height above the ground increases and then decreases again, repeating the same pattern each time the Ferris wheel makes a complete rotation. This pattern is an example of a periodic function. We use periodic functions to model phenomena that exhibit cyclical behavior, such as the height of tides, seasonal patterns of growth in plants and animals, radio waves, and planetary motion.
We’ll create a mathematical model for a ride on a Ferris wheel that has a radius of 100 feet and rotates counterclockwise. Our model will be a function that describes your height above the ground as you ride the wheel. In order to graph the Ferris wheel function, we must first specify the input and output variables, and then choose a coordinate system to display their values.
We’ll place the origin at the center of the Ferris wheel. Then the line from the origin to your position on the wheel makes an angle with the horizontal, as shown at right. This angle, \(\theta\text{,}\) will be the input variable for the function. Your height, \(h\text{,}\) is also a variable, and is related to the \(y\)-coordinate of your position; in fact, we see that \(h = y + 100\text{,}\) because the center of the wheel is 100 feet above the ground.
To simplify the model, we’ll first graph \(y\) as the output variable, instead of \(h\text{.}\) As the angle \(\theta\) increases from \(0\degree\) to \(90\degree\text{,}\) your \(y\)-coordinate increases from 0 to 100. You are then at the top of the wheel. Then, as \(\theta\) increases from \(90\degree\) to \(180\degree\text{,}\) your \(y\)-coordinate decreases from 100 back to 0.
Finally, as \(\theta\) increases from \(180\degree\) to \(360\degree\text{,}\) your \(y\)-coordinate decreases from 0 to \(-100\) and then increases from \(-100\) back to 0. You have made one complete rotation on the Ferris wheel. If you go around again, \(\theta\) increases from \(360\degree\) to \(720\degree\text{,}\) and the graph of your \(y\)-coordinate will repeat the pattern of the first rotation. The figure above shows how your \(y\)-coordinate is plotted as a function of the angle \(\theta\text{.}\)
Look back at the diagram of the Ferris wheel and notice that \(\sin(\theta) = \dfrac{y}{100}\text{,}\) so
\begin{equation*}
y = 100 \sin (\theta)
\end{equation*}
For example, when \(\theta = 30\degree\text{,}\) the \(y\)-coordinate is
\begin{equation*}
y = 100 \sin \left(30\degree\right) = 100\left(\dfrac{1}{2}\right) = 50
\end{equation*}
and your height above the ground is
\begin{equation*}
h = y + 100 = 150~~ \text{feet}
\end{equation*}
In general, then, \(h\) is given as a function of \(\theta\) by
\begin{equation*}
h = y + 100 = 100 \sin \left(\theta\right) + 100
\end{equation*}
This is our model for your height on a Ferris wheel ride.
Subsection The Sine Function
Our Ferris wheel model used values of \(\sin (\theta)\text{,}\) so let us explore its properties. Remember that the trigonometric ratio \(\sin (\theta)\) is actually a function of the angle \(\theta\text{.}\) Thus, for each value of \(\theta\text{,}\) there is only one value of \(\sin (\theta)\text{,}\) and we may write \(f(\theta) = \sin (\theta)\text{.}\)
Activity 4.3. Graph of the Sine Function.
Graph the sine function \(f(\theta) = \sin (\theta)\text{.}\)
-
First make a table of values. You can use your calculator’s table feature to fill in the following values, rounded to two decimal places.
| \(\theta\) |
\(0\degree\) |
\(10\degree\) |
\(20\degree\) |
\(30\degree\) |
\(40\degree\) |
\(50\degree\) |
\(60\degree\) |
\(70\degree\) |
\(80\degree\) |
\(90\degree\) |
| \(\sin (\theta)\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
-
Plot the points in the table and connect them with a smooth curve to graph the sine function for angles in the first quadrant, from \(0\degree\) to \(90\degree\text{.}\)
-
Recall that the values of \(\sin (\theta)\) for angles in the second quadrant can be found using reference angles: \(\sin (180\degree - \theta) = \sin (\theta)\text{.}\) This fact gives us values of \(\sin \theta\) from \(90\degree\) to \(180\degree\text{.}\)
| \(\theta\) |
\(100\degree\) |
\(110\degree\) |
\(120\degree\) |
\(130\degree\) |
\(140\degree\) |
\(150\degree\) |
\(160\degree\) |
\(170\degree\) |
\(180\degree\) |
| \(\sin (\theta)\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
Continue your graph from part (2) for second quadrant angles.
-
The values of \(\sin (\theta)\) in the third and fourth quadrants are the negatives of their values in the second and first quadrants.
| \(\theta\) |
\(190\degree\) |
\(200\degree\) |
\(210\degree\) |
\(220\degree\) |
\(230\degree\) |
\(240\degree\) |
\(250\degree\) |
\(260\degree\) |
\(270\degree\) |
| \(\sin (\theta)\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
| \(\theta\) |
\(280\degree\) |
\(290\degree\) |
\(300\degree\) |
\(310\degree\) |
\(320\degree\) |
\(330\degree\) |
\(340\degree\) |
\(350\degree\) |
\(360\degree\) |
| \(\sin (\theta)\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
\(\hphantom{000}\) |
Continue your graph from part (3) for third and fourth quadrant angles.
-
The graph of \(f(\theta) = \sin (\theta)\) from \(0\degree\) to \(360\degree\) is shown below.
If we continue the graph for angles larger than \(360\degree\) or smaller than \(0\degree\text{,}\) we find that the same pattern repeats, as shown below. This should not be surprising, because we know that coterminal angles have the same trigonometric ratios.
The sine is an example of a periodic function. The smallest interval on which the graph repeats is called the period of the graph. From the graph in the previous example, we make the following observations:
Properties of the Sine Function.
The period of the sine function is \(360\degree\text{.}\)
The maximum and minimum function values are 1 and 1, respectively.
The graph oscillates around its midline, the horizontal line \(y = 0\text{.}\)
The distance between the midline and either the maximum or minimum value is called the amplitude of the function, so the amplitude of the sine function is 1.
You can use your calculator to graph the sine function, by entering
\(\qquad\qquad\qquad \text{Y}_{1} =\)SIN \(\small\boxed{X,T,\theta, n}\) )
and pressing ZOOM \(7\) for the trig window. The graph shows two periods of the sine function, from \(\theta = -360\degree\) to \(\theta = 360\degree\text{.}\)
Checkpoint 4.20.
Your height \(h\) on the Ferris wheel is a function of \(\theta\text{,}\)
\begin{equation*}
h = 100 + y = 100 + 100 \sin (\theta)
\end{equation*}
- Complete the table of values and graph the Ferris wheel function, \(h = F(\theta)\text{.}\)
| \(\theta\) |
\(0\degree\) |
\(30\degree\) |
\(60\degree\) |
\(90\degree\) |
\(120\degree\) |
\(150\degree\) |
\(180\degree\) |
| \(\sin (\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
| \(h = F(\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
| \(\theta\) |
\(210\degree\) |
\(240\degree\) |
\(270\degree\) |
\(300\degree\) |
\(330\degree\) |
\(360\degree\) |
| \(\sin (\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
| \(h = F(\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
Give the period, amplitude, and midline of the graph.
Answer.
-
| \(\theta\) |
\(0\degree\) |
\(30\degree\) |
\(60\degree\) |
\(90\degree\) |
\(120\degree\) |
\(150\degree\) |
\(180\degree\) |
| \(\sin (\theta)\) |
\(0\) |
\(0.5\) |
\(0.866\) |
\(1.0\) |
\(0.866\) |
\(0.5\) |
\(0\) |
| \(h = F(\theta)\) |
\(100\) |
\(150\) |
\(186.6\) |
\(200\) |
\(186.6\) |
\(150\) |
\(100\) |
| \(\theta\) |
\(210\degree\) |
\(240\degree\) |
\(270\degree\) |
\(300\degree\) |
\(330\degree\) |
\(360\degree\) |
| \(\sin (\theta)\) |
\(-0.5\) |
\(-0.866\) |
\(-1.0\) |
\(-0.866\) |
\(-0.5\) |
\(0\) |
| \(h = F(\theta)\) |
\(50\) |
\(13.4\) |
\(0\) |
\(13.4\) |
\(50\) |
\(100\) |
Period: \(360\degree\text{,}\) amplitude: 100, midline: \(h=100\)
Subsection The Cosine Function
In the previous exercise you graphed the height of a person riding on a Ferris wheel. Your graph involved \(\sin (\theta)\text{,}\) because the sine function tells us the \(y\)-coordinate of a point that travels around a circle. The cosine function tells us the \(x\)-coordinate of a point that travels around a circle.
Example 4.21.
Small adjustments to the fit of a bicycle can affect both the cyclist’s efficiency and the stress on his or her jounts. The KOPS rule (Knee Over Pedal Stem) aligns the cyclist’s knee directly over the pedal at the point of maximum force, as shown at right. As the cyclist’s foot rotates away from this KOPS line, stress on the knee increases.
Suppose the pedal crank is 18 centimeters long. When the crank makes an angle \(\theta\) with the horizontal, how far is the cyclist’s foot displaced horizontally from the KOPS line?
Graph the horizontal displacement, \(d\text{,}\) as a function of \(\theta\text{.}\)
Solution.
-
The cyclist’s foot travels around a circle of radius 18 centimeters. If we place the origin at the center of the chain gear, the \(x\)-coordinate of the foot is given by
\begin{equation*}
x = r \cos (\theta) = 18 \cos (\theta)
\end{equation*}
(See the figure at right.) The KOPS line is the vertical line \(x = 18\text{,}\) so the horizontal distance between the cyclist’s foot and the KOPS line is
\begin{equation*}
d = 18 - 18 \cos (\theta)
\end{equation*}
-
You can use your calculator to verify the table and graph for the function
\begin{equation*}
d = 18 - 18 \cos (\theta)
\end{equation*}
shown below.
| \(\theta\) |
\(0\degree\) |
\(30\degree\) |
\(60\degree\) |
\(90\degree\) |
\(120\degree\) |
\(150\degree\) |
\(180\degree\) |
\(210\degree\) |
\(240\degree\) |
\(270\degree\) |
\(300\degree\) |
\(330\degree\) |
\(360\degree\) |
| \(d\) |
\(0\) |
\(2.4\) |
\(9\) |
\(18\) |
\(27\) |
\(33.6\) |
\(36\) |
\(33.6\) |
\(27\) |
\(18\) |
\(9\) |
\(2.4\) |
\(0\) |
In the example above, the period is \(360\degree\text{,}\) the amplitude is 18 cm, and the midline is \(d=18\text{.}\)
You can see that the cosine graph is similar to the sine graph, but they are not identical.
Checkpoint 4.22.
-
Complete the table below with values rounded to two decimal places. Use the table and your knowledge of reference angles to graph the cosine function, \(f(\theta) = \cos (\theta)~\) from \(-180\degree\) to \(540\degree\text{.}\)
| \(\theta\) |
\(0\degree\) |
\(10\degree\) |
\(20\degree\) |
\(30\degree\) |
\(40\degree\) |
\(50\degree\) |
\(60\degree\) |
\(70\degree\) |
\(80\degree\) |
\(90\degree\) |
| \(\cos (\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
Use your graph to find the period, amplitude, and midline of the cosine function. How does the graph of cosine differ from the graph of sine? (Hint: Consider the intercepts of the graph, and the location of the maximum and minimum values.)
Answer.
-
| \(\theta\) |
\(0\degree\) |
\(10\degree\) |
\(20\degree\) |
\(30\degree\) |
\(40\degree\) |
\(50\degree\) |
\(60\degree\) |
\(70\degree\) |
\(80\degree\) |
\(90\degree\) |
| \(\cos (\theta)\) |
\(1\) |
\(0.98\) |
\(0.94\) |
\(0.87\) |
\(0.77\) |
\(0.64\) |
\(0.50\) |
\(0.34\) |
\(0.17\) |
\(0\) |
Period: \(360\degree\text{,}\) amplitude: 1, midline: \(y = 0\text{.}\) The cosine graph starts \((\theta = 0\degree)\) at its high point, while the sine graph starts \((\theta = 0\degree)\) at its midline.
Subsection Interlude: Review of Function Notation
Perhaps it is time to review our use of function notation. Recall that we use the notation \(y = f(x)\) to indicate that \(y\) is a function of \(x\text{,}\) that is, \(x\) is the input variable and \(y\) is the output variable.
Example 4.23.
Make a table of input and output values and a graph for the function \(y = f(x) = \sqrt{9 - x^2}\text{.}\)
Solution.
We choose several values for the input variable, \(x\text{,}\) and evaluate the function to find the corresponding values of the output variable, \(y\text{.}\) For example,
\begin{equation*}
f(-3) = \sqrt{9 - (-3)^2} = 0
\end{equation*}
We plot the points in the table and connect them to obtain the graph shown at right.
| \(x\) |
\(-3\) |
\(-2\) |
\(-1\) |
\(0\) |
\(1\) |
\(2\) |
\(3\) |
| \(y\) |
\(0\) |
\(\sqrt{5}\) |
\(2\sqrt{2}\) |
\(3\) |
\(2\sqrt{2}\) |
\(\sqrt{5}\) |
\(0\) |
Of course, we don’t always use \(x\) and \(y\) for the input and output variables. In the previous example, we could write \(w = f(t) = \sqrt{9 - t^2}\) for the function, so that \(t\) is the input and \(w\) is the output. The table of values and the graph are the same; only the names of the variables have changed.
Checkpoint 4.25.
Sketch a graph of each function, and label the axes.
\(\displaystyle d = F(\phi) = \sin (\phi)\)
\(\displaystyle t = G(\beta) = \cos (\beta)\)
Subsection The Tangent Function
The tangent function is periodic, but its graph is not similar to the graphs of sine and cosine. Recall that the tangent of an angle in standard position is defined by
\begin{equation*}
\tan (\theta) = \dfrac{y}{x}
\end{equation*}
Study the figure at right to see that as \(\theta\) increases from \(0\degree\) to \(90\degree\text{,}\) \(y\) increases while \(x\) remains constant, so the value of \(\tan (\theta)\) increases.
Example 4.27.
Sketch a graph of \(f(\theta) = \tan (\theta)\) for \(0\degree \le \theta \le 180\degree\text{.}\)
Solution.
You can use your calculator to verify the following values for \(\tan \theta\text{.}\)
| \(\theta\) |
\(0\degree\) |
\(30\degree\) |
\(60\degree\) |
\(70\degree\) |
\(80\degree\) |
\(85\degree\) |
| \(\tan (\theta)\) |
\(0\) |
\(0.58\) |
\(1.73\) |
\(2.75\) |
\(5.67\) |
\(11.43\) |
As \(\theta\) gets closer to \(90\degree\text{,}\) \(\tan (\theta)\) increases very rapidly. Recall that \(\tan (90\degree)\) is undefined, so there is no point on the graph at \(\theta = 90\degree\text{.}\) The graph of \(f(\theta) = \tan (\theta)\) for \(0\le\theta\lt 90\degree\) is shown at right.
For \(\theta\) in the second quadrant, the tangent is negative. The reference angle for each angle in the second quadrant is its supplement, so
\begin{equation*}
\tan (\theta) =-\tan (180\degree - \theta)
\end{equation*}
as shown at right. For example, you can verify that
\begin{equation*}
\tan (130\degree) = -\tan (180\degree - 130\degree) = -\tan (50\degree) = -1.19
\end{equation*}
In particular, for values of \(\theta\) close to \(90\degree\text{,}\) the values of \(\tan (\theta)\) are large negative numbers. We plot several points and sketch the graph in the second quadrant.
| \(\theta\) |
\(100\degree\) |
\(110\degree\) |
\(120\degree\) |
\(150\degree\) |
\(180\degree\) |
| \(\tan (\theta)\) |
\(-5.67\) |
\(-2.75\) |
\(-1.73\) |
\(-0.58\) |
\(0\) |
In the figure at right, note that the graph has a break at \(\theta = 90\degree\text{,}\) because \(\tan (90\degree)\) is undefined.
Now let’s consider the graph of \(f(\theta) = \tan (\theta)\) in the third and fourth quadrants. The tangent is positive in the third quadrant, and negative in the fourth quadrant. In fact, from the figure below you can see that the angles \(\theta\) and \(180\degree + \theta\) are vertical angles.
Because \(\theta\) and \(180\degree + \theta\) have the same reference angle, they have the same tangent. For example,
\begin{align*}
\tan (200\degree) \amp = \tan (20\degree)\\
\tan (230\degree) \amp = \tan (50\degree)\\
\tan (250\degree) \amp = \tan (70\degree)
\end{align*}
Thus, the graph of \(\tan (\theta)\) in the third quadrant is the same as its graph in the first quadrant. Similarly, the graph of the tangent function in the fourth quadrant is the same as its graph in the second quadrant. The completed graph is shown below.
Checkpoint 4.28.
What is the period of the tangent function?
Does the graph of tangent have an amplitude?
For what values of \(\theta\) is \(\tan (\theta)\) undefined?
Give the equations of any horizontal or vertical asymptotes for \(0\degree \le \theta \le 360\degree\text{.}\)
Answer.
\(\displaystyle 180\degree\)
No
\(90\degree, ~270\degree\text{,}\) and their coterminal angles
\(\displaystyle \theta = 90\degree, ~\theta = 270\degree\)
Subsection Period, Midline and Amplitude
All sine and cosine graphs have the characteristic "wave" shape we’ve seen in previous examples. But we can alter the size and frequency of the waves by changing the formula for the function. In the next example we consider three variations of the sine function.
Example 4.29.
Make a table of values and sketch a graph for each of the functions. How does each differ from the graph of \(y = \sin (\theta)\) ?
\(\displaystyle y = 3\sin (\theta)\)
\(\displaystyle y = 3 + \sin (\theta)\)
\(\displaystyle y = \sin (3\theta)\)
Solution.
-
We make a table with multiples of \(45\degree\text{.}\)
| \(\theta\) |
\(0\degree\) |
\(45\degree\) |
\(90\degree\) |
\(135\degree\) |
\(180\degree\) |
\(225\degree\) |
\(270\degree\) |
\(315\degree\) |
\(360\degree\) |
| \(y = 3\sin (\theta)\) |
\(0\) |
\(2.1\) |
\(3\) |
\(2.1\) |
\(0\) |
\(-2.1\) |
\(-3\) |
\(-2.1\) |
\(0\) |
We plot the points, and connect them with a sine-shaped wave. Compare the graph, shown at right, to the graph of \(y = \sin (\theta)\text{.}\) The graph is like a sine graph, except that it oscillates between a maximum value of \(3\) and a minimum value of \(-3\text{.}\) The amplitude of this function is 3.
-
Again, we make a table of values with multiples of \(45\degree\) and plot the points.
| \(\theta\) |
\(0\degree\) |
\(45\degree\) |
\(90\degree\) |
\(135\degree\) |
\(180\degree\) |
\(225\degree\) |
\(270\degree\) |
\(315\degree\) |
\(360\degree\) |
| \(y = 3 + \sin (\theta)\) |
\(3\) |
\(3.7\) |
\(4\) |
\(3.7\) |
\(3\) |
\(2.3\) |
\(2\) |
\(2.1\) |
\(3\) |
This graph has the same amplitude as \(y = \sin (\theta)\text{,}\) but the entire graph is shifted up by 3 units, as shown at right. The midline of this function is the line \(y = 3\text{.}\)
-
This time we’ll make a table with multiples of \(15\degree\text{.}\)
| \(\theta\) |
\(15\degree\) |
\(30\degree\) |
\(45\degree\) |
\(60\degree\) |
\(75\degree\) |
\(90\degree\) |
\(105\degree\) |
\(120\degree\) |
\(135\degree\) |
\(150\degree\) |
\(165\degree\) |
\(180\degree\) |
| \(y = \sin (3\theta)\) |
\(0.7\) |
\(1\) |
\(0.7\) |
\(0\) |
\(-0.7\) |
\(-1\) |
\(-0.7\) |
\(0\) |
\(0.7\) |
\(1\) |
\(0.7\) |
\(0\) |
You can continue the table for \(\theta\) between \(180\degree\) and \(360\degree\text{,}\) and plot the points to find the graph shown at right. The graph has the same amplitude and midline as \(y = \sin (\theta)\text{,}\) but it completes three cycles from \(0\degree\) to \(360\degree\) instead of one cycle. The period of this graph is one-third of \(360\degree\text{,}\) or \(120\degree\text{.}\)
The graphs in the previous example illustrate a general rule about sine and cosine graphs.
Amplitude, Period, and Midline.
The graph of
\begin{equation*}
\blert{y = A\cos(\theta)} ~~~\text{or}~~~ \blert{y = A\sin(\theta)}
\end{equation*}
has amplitude
\(\abs{A}\text{.}\)
The graph of
\begin{equation*}
\blert{y =\cos (B\theta)} ~~~\text{or}~~~ \blert{y = \sin 9}
\end{equation*}
has period
\(\dfrac{360\degree}{\abs{B}}\text{.}\)
The graph of
\begin{equation*}
\blert{y = k + \cos(\theta)} ~~~\text{or}~~~ \blert{y =k + \sin(\theta)}
\end{equation*}
has midline
\(y = k\text{.}\)
Checkpoint 4.30.
Sketch a graph for each of the following functions. Describe how each is different from the graph of \(y = \cos \theta\text{.}\)
\(\displaystyle h(\theta) = 2\cos (\theta)\)
\(\displaystyle g(\theta) = \cos (2\theta)\)
\(\displaystyle f(\theta) = 2 + \cos (\theta)\)
Answer.
The period is \(180\degree\text{.}\)
The midline is \(y = 2\text{.}\)
The quantities \(A, B,\) and \(k\) in the equations above are called parameters, and their values for a particular function give us information about its graph.
Example 4.31.
State the period, midline, and amplitude of the graph of \(y = -3 + 4\sin (3\theta),\) and graph the function.
Solution.
For this function, \(A = 4,~ B = 3,\) and \(k = -3\text{.}\) Its amplitude is 4, its period is \(\dfrac{360\degree}{3} = 120\degree\text{,}\) and its midline is \(y = -3\text{.}\) The graph is shown at right.
Checkpoint 4.32.
State the period, midline, and amplitude of the graph of \(y = 1 - 3\sin (2\theta),\) and graph the function.
Answer.
Amplitude 3, period \(180\degree\text{,}\) midline \(y = 1\)
Subsection
Review the following skills you will need for this section.
Algebra Refresher 4.4.
Graph the function.
Give the coordinates of any intercepts, and any maximum or minimum values.
\(\displaystyle f(x) = -6 + \dfrac{2}{3} x\)
\(\displaystyle g(x) = 4 - \dfrac{3}{2} x\)
\(\displaystyle p(t) = t^2 - 4\)
\(\displaystyle q(t) = 9 - t^2\)
\(\displaystyle F(x) = 2 - \sqrt{z}\)
\(\displaystyle G(z) = \sqrt{4 - z}\)
\(\underline{\qquad\qquad\qquad\qquad}\)
Algebra Refresher Answers
\((0,-6)\text{,}\) \(~ (9,0)\)
\((0,4)\text{,}\) \(~ \left(\dfrac{8}{3},0\right)\)
\((0,-4)\text{,}\) \(~ (-2,0)\text{,}\) \(~ (2,0)\text{,}\) Min: \(-4\)
\((0,9)\text{,}\) \(~ (-3,0)\text{,}\) \(~ (3,0)\text{,}\) \(\text{Max: }9\)
\((0,2)\text{,}\) \(~ (4,0)\text{,}\) Max: \(2\)
\((0,2)\text{,}\) \(~ (4,0)\text{,}\) Min: \(~0\)
Subsection Section 4.2 Summary
Subsubsection Vocabulary
Input variable
Output variable
Periodic function
Period
Midline
Amplitude
Asymptote
Angle of Inclination
Bearings
Subsubsection Concepts
Coordinates.
If point \(P\) is located at a distance \(r\) from the origin in the direction specified by angle \(\theta\) in standard position, then the coordinates of \(P\) are
\begin{equation*}
x = r \cos (\theta) ~~~~ \text{and} ~~~~ y = r \sin (\theta)
\end{equation*}
Navigational directions for ships and planes are sometimes given as bearings, which are angles measured clockwise from north.
Periodic functions are used to model phenomena that exhibit cyclical behavior.
The trigonometric ratios \(\sin (\theta)\) and \(\cos (\theta)\) are functions of the angle \(\theta\text{.}\)
The period of the sine function is \(360\degree\text{.}\) Its midline is the horizontal line \(y = 0\text{,}\) and the amplitude of the sine function is 1.
The graph of the cosine function has the same period, midline, and amplitude as the graph of the sine function. However, the locations of the intercepts and of the maximum and minimum values are different.
We use the notation \(y = f(x)\) to indicate that \(y\) is a function of \(x\text{,}\) that is, \(x\) is the input variable and \(y\) is the output variable.
The tangent function has period \(180\degree\text{.}\) It is undefined at odd multiples of \(90\degree\text{,}\) and is increasing on each interval of its domain.
Angle of Inclination.
The angle of inclination of a line is the angle \(\alpha\) measured in the positive direction from the positive \(x\)-axis to the line. If the slope of the line is \(m\text{,}\) then
\begin{equation*}
\tan \alpha = m
\end{equation*}
where \(0\degree \le \alpha \le 180\degree\text{.}\)
Subsubsection Study Questions
-
Use the figure to help you fill in the blanks.
As \(\theta\) increases from \(0\degree\) to \(90\degree\text{,}\) \(f(\theta) = \sin (\theta)\) from to .
As \(\theta\) increases from \(90\degree\) to \(180\degree\text{,}\) \(f(\theta) = \sin (\theta)\) from to .
As \(\theta\) increases from \(180\degree\) to \(270\degree\text{,}\) \(f(\theta) = \sin (\theta)\) from to .
As \(\theta\) increases from \(270\degree\) to \(360\degree\text{,}\) \(f(\theta) = \sin (\theta)\) from to .
-
Use the figure to help you fill in the blanks.
As \(\theta\) increases from \(0\degree\) to \(90\degree\text{,}\) \(f(\theta) = \cos (\theta)\) from to .
As \(\theta\) increases from \(90\degree\) to \(180\degree\text{,}\) \(f(\theta) = \cos (\theta)\) from to .
As \(\theta\) increases from \(180\degree\) to \(270\degree\text{,}\) \(f(\theta) = \cos (\theta)\) from to .
As \(\theta\) increases from \(270\degree\) to \(360\degree\text{,}\) \(f(\theta) = \cos (\theta)\) from to .
List several ways in which the graph of \(y = \tan \theta\) is different from the graphs of \(y = \sin (\theta)\) and \(y = \cos (\theta)\text{.}\)
If the angle of inclination of a line is greater than \(45\degree\text{,}\) what can you say about its slope?
Subsubsection Skills
Find coordinates #1-12, 19-20
Use bearings to determine position #13-18
Sketch graphs of the sine and cosine functions #21-26, 31-32
Find the coordinates of points on a sine or cosine graph #27-30, 41-44
Use function notation #33-40
Find reference angles #45-48
Solve equations graphically #49-56
Graph the tangent function #57-60
Find and use the angle of inclination of a line #61-70
Exercises Homework 4.2
1.
Prepare a graph with the horizontal axis scaled from \(0\degree\) to \(360\degree\) in multiples of \(45\degree\text{.}\)
Sketch a graph of \(f(\theta) = \sin (\theta)\) by plotting points for multiples of \(45\degree\text{.}\)
2.
Prepare a graph with the horizontal axis scaled from \(0\degree\) to \(360\degree\) in multiples of \(45\degree\text{.}\)
Sketch a graph of \(f(\theta) = \cos (\theta)\) by plotting points for multiples of \(45\degree\text{.}\)
3.
Prepare a graph with the horizontal axis scaled from \(0\degree\) to \(360\degree\) in multiples of \(30\degree\text{.}\)
Sketch a graph of \(f(\theta) = \cos (\theta)\) by plotting points for multiples of \(30\degree\text{.}\)
4.
Prepare a graph with the horizontal axis scaled from \(0\degree\) to \(360\degree\) in multiples of \(30\degree\text{.}\)
Sketch a graph of \(f(\theta) = \sin (\theta)\) by plotting points for multiples of \(30\degree\text{.}\)
Exercise Group.
For Problems 5–8, give the coordinates of each point on the graph of \(f(\theta) = \sin (\theta)\) or \(f(\theta) = \cos (\theta)\text{.}\)
9.
Make a short table of values like the one shown, and sketch the function by hand. Be sure to label the \(x\)-axis and \(y\)-axis appropriately.
| \(\theta\) |
\(0\degree\) |
\(90\degree\) |
\(180\degree\) |
\(270\degree\) |
\(360\degree\) |
| \(f(\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\displaystyle f(\theta) = \sin (\theta)\)
\(\displaystyle f(\theta) = \cos (\theta)\)
10.
One of these graphs is \(y = A \sin (k\theta)\text{,}\) and the other is \(y = A \cos (k\theta)\text{.}\) Explain how you know which is which.
Exercise Group.
For Problems 11–18, evaluate the expression for \(f(\theta) = \sin (\theta)\) and \(g(\theta) = \cos (\theta)\text{.}\)
11.
\(3 + f(30\degree)\)
12.
\(3 f(30\degree)\)
13.
\(4g(225\degree) - 1\)
14.
\(-4 + 2g(225\degree) - 1\)
15.
\(-2f(3\theta)\text{,}\) for \(\theta = 90\degree\)
16.
\(6f\left(\dfrac{\theta}{2}\right)\text{,}\) for \(\theta = 90\degree\)
17.
\(8 - 5g\left(\dfrac{\theta}{3}\right)\text{,}\) for \(\theta = 360\degree\)
18.
\(1 - 4g(4\theta)\text{,}\) for \(\theta = 135\degree\)
19.
The graph shows your height as a function of angle as you ride the Ferris wheel. For each location \(A\)–\(E\) on the Ferris wheel, mark the corresponding point on the graph.
20.
The graph shows your height as a function of angle as you ride the Ferris wheel. For each location \(F\)–\(J\) on the graph, mark the corresponding point on the Ferris wheel.
21.
The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location \(K\)–\(O\) on the chain gear, mark the corresponding point on the graph.
22.
The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location \(P\)–\(T\) on the graph, mark the corresponding point on the chain gear.
23.
-
Fill in the table for values of \(\tan (\theta)\text{.}\) Round your answers to three decimal places.
| \(\theta\) |
\(81\degree\) |
\(82\degree\) |
\(83\degree\) |
\(84\degree\) |
\(85\degree\) |
\(86\degree\) |
\(87\degree\) |
\(88\degree\) |
\(89\degree\) |
| \(\tan (\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
What happens to \(\tan (\theta)\) as \(\theta\) increases toward \(90\degree\text{?}\)
-
Fill in the table for values of \(\tan (\theta)\text{.}\) Round your answers to three decimal places.
| \(\theta\) |
\(99\degree\) |
\(98\degree\) |
\(97\degree\) |
\(96\degree\) |
\(95\degree\) |
\(94\degree\) |
\(93\degree\) |
\(92\degree\) |
\(91\degree\) |
| \(\tan (\theta)\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
What happens to \(\tan (\theta)\) as \(\theta\) decreases toward \(90\degree\text{?}\)
What value does your calculator give for \(\tan (90\degree)\text{?}\) Why?
24.
-
Fill in the table with exact values of \(\tan (\theta)\text{.}\) Then give decimal approximations to two places.
| \(\theta\) |
\(0\degree\) |
\(30\degree\) |
\(45\degree\) |
\(60\degree\) |
\(90\degree\) |
\(120\degree\) |
\(135\degree\) |
\(150\degree\) |
\(180\degree\) |
|
\(\tan (\theta)\) (exact) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
|
\(\tan (\theta)\) (approx.) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
-
Fill in the table with exact values of \(\tan (\theta)\text{.}\) Then give decimal approximations to two places.
| \(\theta\) |
\(180\degree\) |
\(210\degree\) |
\(225\degree\) |
\(240\degree\) |
\(270\degree\) |
\(300\degree\) |
\(315\degree\) |
\(330\degree\) |
\(360\degree\) |
|
\(\tan (\theta)\) (exact) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
|
\(\tan (\theta)\) (approx.) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
-
Plot the points from the tables and sketch a graph of \(f(\theta) = \tan (\theta)\text{.}\)
25.
Write an equation for a sine function with amplitude 6.
26.
Write an equation for a cosine function with amplitude \(\dfrac{1}{2}\text{.}\)
27.
Write an equation for a cosine function with midline \(-5\text{.}\)
28.
Write an equation for a sine function with midline 2.
29.
Write an equation for a sine function with period \(90\degree\text{.}\)
30.
Write an equation for a cosine function with period \(720\degree\text{.}\)
Exercise Group.
For Problems 31–36,
Graph the function.
State the amplitude, period and midline of the function.
31.
\(y = 3 \cos (\theta)\)
32.
\(y = 4 \sin (\theta)\)
33.
\(y = 3 + \sin (\theta)\)
34.
\(y = -2 + \cos (\theta)\)
35.
\(y = \cos (3\theta)\)
36.
\(y = \sin (2 \theta)\)
Exercise Group.
For Problems 37–42, give the coordinates of the points on the graph.
37.
\(f(\theta) = -3 \cos (\theta)\)
38.
\(f(\theta) = -4 \sin (\theta)\)
39.
\(f(\theta) =\sin (4\theta)\)
40.
\(f(\theta) = -\cos (3\theta)\)
41.
\(f(\theta) = -3 + \cos (\theta)\)
42.
\(f(\theta) = 1 + \sin (\theta)\)
Exercise Group.
For Problems 43–48, graph the function using technology. State the amplitude, period, and midline.
43.
\(y = 3 + 4 \cos (\theta)\)
44.
\(y = -4 + 3 \sin (\theta)\)
45.
\(y = 5 \sin (2\theta)\)
46.
\(y = 6 \cos (4\theta)\)
47.
\(f(\theta) = -4 + 3 \sin (3\theta)\)
48.
\(f(\theta) = 2 + 4 \cos (3\theta)\)
Exercise Group.
For Problems 49–56,
State the amplitude, period, and midline for the graph.
Write an equation for the graph using sine or cosine.
49.
50.
51.
52.
53.
54.
55.
56.
Exercise Group.
For Problems 57–62, write the equation of a sine or cosine function with the given properties.
57.
Midline \(y = -4\text{,}\) amplitude \(6\text{,}\) period \(120\degree\)
58.
Midline \(y = 5\text{,}\) amplitude \(\dfrac{3}{2}\text{,}\) period \(180\degree\)
59.
Maximum points at \((0\degree, 5)\) and \((360\degree, 5)\text{,}\) minimum point at \((180\degree, 1)\)
60.
Maximum point at \((90\degree, 1)\text{,}\) minimum point at \((270\degree, -3)\)
61.
Horizontal intercepts at \(45\degree\) and \(135\degree\text{,}\) vertical intercept at \((0\degree, 12)\)
62.
Horizontal intercepts at \(30\degree\) and \(90\degree\text{,}\) vertical intercept at \((0\degree,-8)\)
Exercise Group.
For Problems 63–66, the table describes a sine or cosine function. Find an equation for the function.
63.
| \(\theta\) |
\(0\degree\) |
\(45\degree\) |
\(90\degree\) |
\(135\degree\) |
\(180\degree\) |
\(225\degree\) |
\(270\degree\) |
\(315\degree\) |
\(360\degree\) |
| \(f(\theta)\) |
\(7\) |
\(5.56\) |
\(2\) |
\(-1.54\) |
\(-3\) |
\(-1.54\) |
\(2\) |
\(5.54\) |
\(7\) |
64.
| \(\theta\) |
\(0\degree\) |
\(45\degree\) |
\(90\degree\) |
\(135\degree\) |
\(180\degree\) |
\(225\degree\) |
\(270\degree\) |
\(315\degree\) |
\(360\degree\) |
| \(f(\theta)\) |
\(1\) |
\(3.12\) |
\(4\) |
\(3.12\) |
\(1\) |
\(-1.12\) |
\(-2\) |
\(-1.12\) |
\(1\) |
65.
| \(\theta\) |
\(0\degree\) |
\(45\degree\) |
\(90\degree\) |
\(135\degree\) |
\(180\degree\) |
\(225\degree\) |
\(270\degree\) |
\(315\degree\) |
\(360\degree\) |
| \(f(\theta)\) |
\(0\) |
\(-2.83\) |
\(-4\) |
\(-2.83\) |
\(0\) |
\(2.83\) |
\(4\) |
\(2.83\) |
\(0\) |
66.
| \(\theta\) |
\(0\degree\) |
\(45\degree\) |
\(90\degree\) |
\(135\degree\) |
\(180\degree\) |
\(225\degree\) |
\(270\degree\) |
\(315\degree\) |
\(360\degree\) |
| \(f(\theta)\) |
\(-9\) |
\(-6.36\) |
\(0\) |
\(6.36\) |
\(9\) |
\(6.36\) |
\(0\) |
\(-6.36\) |
\(-9\) |
