Chapter 4: Trig Functions
4.2 Graphs of Trigonometric Functions
Algebra Refresher
Graph the function, the coordinates of any intercepts, and any maximum or minimum values.
[latex]\displaystyle f(x) = -6 + \dfrac{2}{3} x[/latex]
[latex]\displaystyle g(x) = 4 - \dfrac{3}{2} x[/latex]
[latex]\displaystyle p(t) = t^2 - 4[/latex]
[latex]\displaystyle q(t) = 9 - t^2[/latex]
[latex]\displaystyle F(x) = 2 - \sqrt{z}[/latex]
[latex]\displaystyle G(z) = \sqrt{4 - z}[/latex]
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
Algebra Refresher Answers
[latex]\displaystyle (0,-6), ~ (9,0)[/latex]
[latex](0,4){,}[/latex] [latex]~ \left(\dfrac{8}{3},0\right)[/latex]
[latex]\displaystyle (0,-4), ~ (-2,0), ~ (2,0), ~ {Min:}~-4[/latex]
[latex]\displaystyle (0,9), ~ (-3,0), ~ (3,0), ~ {Max:}~9[/latex]
[latex]\displaystyle (0,2), ~ (4,0), ~ {Max:}~2[/latex]
[latex]\displaystyle (0,2), ~ (4,0), ~ {Min:}~0[/latex]
Learning Objectives
- Find coordinates
- Use bearings to determine position
- Sketch graphs of the sine and cosine functions
- Find the coordinates of points on a sine or cosine graph
- Use function notation
- Find reference angles
- Solve equations graphically
- Graph the tangent function
- Find and use the angle of inclination of a line
Location by Coordinates
One of the most useful applications of the trigonometric ratios allows us to find distances or locations specified by angles. Starting with the definitions of sine and cosine,
[latex]\cos \theta = \dfrac{x}{r} ~~~~ {and} ~~~~ \sin \theta = \dfrac{y}{r}[/latex]
we can solve for [latex]x[/latex] and [latex]y{,}[/latex] the coordinates of points on the terminal side of the angle, and obtain the following results.
Coordinates.
If point [latex]P[/latex] is located at a distance [latex]r[/latex] from the origin in the direction specified by angle [latex]\theta[/latex] in standard position, then the coordinates of [latex]P[/latex] are
[latex]{x = r \cos \theta ~~~~ {and} ~~~~ y = r \sin \theta}[/latex]
Example 4.25.
Solution
The location of point [latex]P[/latex] is shown at right. We see that [latex]r = 6{,}[/latex] and we can use a calculator to evaluate [latex]\cos 292°[/latex] and [latex]\sin 292°{.}[/latex]
[latex]x = r \cos 292° {and} y = r \sin 292°\ = 6(0.3746) = 6(-0.9272)\ = 2.2476 = -5.5632[/latex]
The coordinates of [latex]P[/latex] are approximately [latex](2.25, -5.56){.}[/latex]
Checkpoint 4.26.
- Find the cosine of [latex]215°{.}[/latex] How far west should you walk from the big oak in order to be directly north of the treasure?
- Find the tangent of [latex]215°{.}[/latex] How far south should you walk from your present location before you begin digging?
Solution
- 409.58 yds
- 286.79 yds
Bearings
Navigational directions for ships and planes are sometimes given as bearings, which are angles measured clockwise from north. For example, a bearing of [latex]110°[/latex] is equivalent to an angle of [latex]20°[/latex] in standard position, or to its coterminal angle [latex]340°{,}[/latex] as shown at right.
From this example, we see that to convert a bearing to an angle [latex]\theta[/latex] in standard position, we can subtract the bearing from [latex]90°{,}[/latex] or
[latex]{\theta = -\text{bearing} + 90°}[/latex]
Example 4.27.
Francine leaves the airport at a bearing of [latex]245°[/latex] and flies 60 miles. How far south of the airport is she at that time?
Solution
A bearing of [latex]245°[/latex] is in the same direction as an angle of
[latex]-245° + 90° = -155°[/latex]
in standard position, as shown at right, or as the coterminal angle
[latex]-155° + 360° = 205°{.}[/latex]
We would like the [latex]y[/latex]-coordinate of Francine’s position, so we calculate [latex]y = r \sin 205°{.}[/latex]
[latex]y = r \sin 205°\ = 60(-0.4226) = -25.36[/latex]
Francine is about 25.4 miles south of the airport.
Checkpoint 4.32.
- 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, [latex]f(\theta) = \cos \theta~[/latex] from [latex]-180°[/latex] to [latex]540°{.}[/latex]
[latex]\theta[/latex] [latex]0°[/latex] [latex]10°[/latex] [latex]20°[/latex] [latex]30°[/latex] [latex]40°[/latex] [latex]50°[/latex] [latex]60°[/latex] [latex]70°[/latex] [latex]80°[/latex] [latex]90°[/latex] [latex]\cos \theta[/latex] - 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.)
Solution
-
[latex]\theta[/latex] [latex]0°[/latex] [latex]10°[/latex] [latex]20°[/latex] [latex]30°[/latex] [latex]40°[/latex] [latex]50°[/latex] [latex]60°[/latex] [latex]70°[/latex] [latex]80°[/latex] [latex]90°[/latex] [latex]\cos \theta[/latex] [latex]1[/latex] [latex]0.98[/latex] [latex]0.94[/latex] [latex]0.87[/latex] [latex]0.77[/latex] [latex]0.64[/latex] [latex]0.50[/latex] [latex]0.34[/latex] [latex]0.17[/latex] [latex]0[/latex] - Period: [latex]360°[/latex]; amplitude: 1; midline: [latex]y = 0{.}[/latex] The cosine graph starts [latex](\theta = 0°)[/latex] at its high point, while the sine graph starts [latex](\theta = 0°)[/latex] at its midline.
Function Notation
We use the notation [latex]y = f(x)[/latex] to indicate that [latex]y[/latex] is a function of [latex]x[/latex]; that is, [latex]x[/latex] is the input variable and [latex]y[/latex] is the output variable.
Example 4.33.
Solution
We choose several values for the input variable, [latex]x{,}[/latex] and evaluate the function to find the corresponding values of the output variable, [latex]y{.}[/latex] For example,
[latex]\begin{equation*} f(-3) = \sqrt{9 - (-3)^2} = 0 \end{equation*}[/latex]
We plot the points in the table and connect them to obtain the graph shown at right.
| [latex]x[/latex] | [latex]-3[/latex] | [latex]-2[/latex] | [latex]-1[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] | [latex]3[/latex] |
| [latex]y[/latex] | [latex]0[/latex] | [latex]\sqrt{5}[/latex] | [latex]2\sqrt{2}[/latex] | [latex]3[/latex] | [latex]2\sqrt{2}[/latex] | [latex]\sqrt{5}[/latex] | [latex]0[/latex] |
Of course, we don’t always use [latex]x[/latex] and [latex]y[/latex] for the input and output variables. In the previous example, we could write [latex]w = f(t) = \sqrt{9 - t^2}[/latex] for the function so that [latex]t[/latex] is the input and [latex]w[/latex] is the output. The table of values and the graph are the same; only the names of the variables have changed.
Caution 4.34.
When we discuss trigonometric functions, there are several variables involved. Our definitions of the trig ratios involve four variables: [latex]x{,}[/latex] [latex]y{,}[/latex] [latex]r{,}[/latex] and [latex]\theta{,}[/latex] as illustrated below.
If the value of [latex]r[/latex] is fixed for a given situation, such as the Ferris wheel or the bicycle wheel discussed above, then [latex]x[/latex] and [latex]y[/latex] are both functions of [latex]\theta{.}[/latex] This means that the values of [latex]x[/latex] and [latex]y[/latex] depend only on the value of the angle [latex]\theta{.}[/latex] If [latex]r = 1{,}[/latex] we have
[latex]\begin{align*} x = f(\theta) = \cos \theta\\ y = g(\theta) = \sin \theta \end{align*}[/latex]
The graphs of these functions are shown below. Note particularly that the horizontal axis displays values of the input variable, and the vertical axis displays the output variable.
If we use different variables for the input and output, the functions and their graphs are the same, but the axes should be labeled with the appropriate variables.
Checkpoint 4.35.
- [latex]\displaystyle d = F(\phi) = \sin \phi[/latex]
- [latex]\displaystyle t = G(\beta) = \cos \beta[/latex]
Solution
Caution 4.36.
When we write [latex]\cos \theta{,}[/latex] we really mean [latex]\cos (\theta){,}[/latex] because we are using “cos” as the name of a function whose input is [latex]\theta{.}[/latex]
A common mistake is to think of [latex]\cos \theta[/latex] or [latex]\cos (\theta)[/latex] as a product, cos times [latex]\theta{,}[/latex] but this makes no sense, because “cos” by itself has no meaning. Remember that [latex]\cos \theta[/latex] represents a single number; namely, the output of the cosine function.
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
[latex]\begin{equation*} \tan \theta = \dfrac{y}{x} \end{equation*}[/latex]
Study the figure at right to see that as [latex]\theta[/latex] increases from [latex]0°[/latex] to [latex]90°{,}[/latex] [latex]y[/latex] increases while [latex]x[/latex] remains constant, so the value of [latex]\tan \theta[/latex] increases.
Example 4.37.
Solution
You can use your calculator to verify the following values for [latex]\tan \theta{.}[/latex]
| [latex]\theta[/latex] | [latex]0°[/latex] | [latex]30°[/latex] | [latex]60°[/latex] | [latex]70°[/latex] | [latex]80°[/latex] | [latex]85°[/latex] |
| [latex]\tan \theta[/latex] | [latex]0[/latex] | [latex]0.58[/latex] | [latex]1.73[/latex] | [latex]2.75[/latex] | [latex]5.67[/latex] | [latex]11.43[/latex] |
As [latex]\theta[/latex] gets closer to [latex]90°{,}[/latex] [latex]\tan \theta[/latex] increases very rapidly. Recall that [latex]\tan 90°[/latex] is undefined, so there is no point on the graph at [latex]\theta = 90°{.}[/latex] The graph of [latex]f(\theta) = \tan \theta[/latex] in the first quadrant is shown at right.
In the second quadrant, the tangent is negative. The reference angle for each angle in the second quadrant is its supplement, so
[latex]\begin{equation*} \tan \theta =-\tan (180° - \theta) \end{equation*}[/latex]
as shown at right. For example, you can verify that
[latex]\begin{equation*} \tan 130° = -\tan (180° - 130°) = -\tan 50° = -1.19 \end{equation*}[/latex]
In particular, for values of [latex]\theta[/latex] close to [latex]90°{,}[/latex] the values of [latex]\tan \theta[/latex] are large negative numbers. We plot several points and sketch the graph in the second quadrant.
| [latex]\theta[/latex] | [latex]100°[/latex] | [latex]110°[/latex] | [latex]120°[/latex] | [latex]150°[/latex] | [latex]180°[/latex] |
| [latex]\tan \theta[/latex] | [latex]-5.67[/latex] | [latex]-2.75[/latex] | [latex]-1.73[/latex] | [latex]-0.58[/latex] | [latex]0[/latex] |
In the figure at right, note that the graph has a break at [latex]\theta = 90°{,}[/latex] because [latex]\tan 90°[/latex] is undefined.
Now let’s consider the graph of [latex]f(\theta) = \tan \theta[/latex] 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 [latex]\theta[/latex] and [latex]180° + \theta[/latex] are vertical angles.
Because [latex]\theta[/latex] and [latex]180° + \theta[/latex] have the same reference angle, they have the same tangent. For example,
[latex]\begin{align*} \tan 200° = \tan 20°\\ \tan 230° = \tan 50°\\ \tan 250° = \tan 70° \end{align*}[/latex]
Thus, the graph of [latex]\tan \theta[/latex] 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 4.38.
- What is the period of the tangent function?
- Does the graph of tangent have an amplitude?
- For what values of [latex]\theta[/latex] is [latex]\tan \theta[/latex] undefined?
- Give the equations of any horizontal or vertical asymptotes for [latex]0° \le \theta \le 360°{.}[/latex]
Solution
- [latex]\displaystyle 180°[/latex]
- No
- [latex]90°, ~270°{,}[/latex] and their coterminal angles
- [latex]\displaystyle \theta = 90°, ~\theta = 270°[/latex]
Angle of Inclination
The figure at right shows a line in the [latex]xy[/latex]-plane. The angle [latex]\alpha[/latex] measured in the positive direction from the positive [latex]x[/latex]-axis to the line is called the angle of inclination of the line.
Note 4.39.
Recall that the slope of a line is given by the ratio [latex]\dfrac {{change in}~ y}{{change in}~x}[/latex] as we move from one point to another on the line. So if we create a right triangle by dropping a perpendicular segment from the line to the [latex]x[/latex]-axis, the ratio of sides [latex]\dfrac {{opposite}}{{adjacent}}[/latex] gives the slope of the line.
Angle of Inclination.
The angle of inclination of a line is the angle [latex]\alpha[/latex] measured in the positive direction from the positive [latex]x[/latex]-axis to the line. If the slope of the line is [latex]m{,}[/latex] then
[latex]\begin{equation*} {\tan \alpha = m} \end{equation*}[/latex]
where [latex]0° \le \alpha \le 180°{.}[/latex]
Example 4.40.
Find the angle of inclination of the line [latex]y = \frac{3}{4}x - 3{.}[/latex]
Solution
The slope of the line is [latex]\frac{3}{4}{.}[/latex] Therefore,
[latex]\tan \alpha = \dfrac{3}{4}\[latex]/latex] [latex]\alpha = \tan^{-1} \left(\dfrac{3}{4}\right) = 36.9°[/latex]
The angle of inclination is [latex]36.9°{.}[/latex]
Checkpoint 4.41.
Find the angle of inclination of the line shown at right,
[latex]y = \dfrac{-6}{5}x + 2[/latex]
Solution
[latex]129.8°[/latex]
Section 4.2 Summary
Vocabulary
- Input variable
- Output variable
- Periodic function
- Period
- Midline
- Amplitude
- Asymptote
- Angle of Inclination
- Bearings
Concepts
-
-
Coordinates.
If point [latex]P[/latex] is located at a distance [latex]r[/latex] from the origin in the direction specified by angle [latex]\theta[/latex] in standard position, then the coordinates of [latex]P[/latex] are
[latex]\begin{equation*} x = r \cos \theta ~~~~ {and} ~~~~ y = r \sin \theta \end{equation*}[/latex]
- 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 [latex]\sin \theta[/latex] and [latex]\cos \theta[/latex] are functions of the angle [latex]\theta{.}[/latex]
- The period of the sine function is [latex]360°{.}[/latex] Its midline is the horizontal line [latex]y = 0{,}[/latex] 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 [latex]y = f(x)[/latex] to indicate that [latex]y[/latex] is a function of [latex]x[/latex]; that is, [latex]x[/latex] is the input variable and [latex]y[/latex] is the output variable.
- The tangent function has period [latex]180°{.}[/latex] It is undefined at odd multiples of [latex]90°[/latex] and is increasing on each interval of its domain.
-
Angle of Inclination.
The angle of inclination of a line is the angle [latex]\alpha[/latex] measured in the positive direction from the positive [latex]x[/latex]-axis to the line. If the slope of the line is [latex]m{,}[/latex] then
[latex]\begin{equation*} \tan \alpha = m \end{equation*}[/latex]
where [latex]0° \le \alpha \le 180°{.}[/latex]
-
Study Questions
- Use the figure to help you fill in the blanks.
- As [latex]\theta[/latex] increases from [latex]0°[/latex] to [latex]90°{,}[/latex] [latex]f(\theta) = \sin \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- As [latex]\theta[/latex] increases from [latex]90°[/latex] to [latex]180°{,}[/latex] [latex]f(\theta) = \sin \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- As [latex]\theta[/latex] increases from [latex]180°[/latex] to [latex]270°{,}[/latex] [latex]f(\theta) = \sin \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- As [latex]\theta[/latex] increases from [latex]270°[/latex] to [latex]360°{,}[/latex] [latex]f(\theta) = \sin \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- Use the figure to help you fill in the blanks.
- As [latex]\theta[/latex] increases from [latex]0°[/latex] to [latex]90°{,}[/latex] [latex]f(\theta) = \cos \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- As [latex]\theta[/latex] increases from [latex]90°[/latex] to [latex]180°{,}[/latex] [latex]f(\theta) = \cos \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- As [latex]\theta[/latex] increases from [latex]180°[/latex] to [latex]270°{,}[/latex] [latex]f(\theta) = \cos \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- As [latex]\theta[/latex] increases from [latex]270°[/latex] to [latex]360°{,}[/latex] [latex]f(\theta) = \cos \theta[/latex] from [latex]\underline\qquad[/latex] to [latex]\underline\qquad[/latex].
- List several ways in which the graph of [latex]y = \tan \theta[/latex] is different from the graphs of [latex]y = \sin \theta[/latex] and [latex]y = \cos \theta{.}[/latex]
- If the angle of inclination of a line is greater than [latex]45°{,}[/latex] what can you say about its slope?