3. Graphing Trigonometric Functions
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Chapter 6
3.
Graphing Trigonometric Functions
This lesson focuses on teaching how to graph trigonometric functions like sine and cosine. It breaks down the process into understandable steps, emphasizing the importance of key elements such as amplitude, period, and frequency. The lesson is particularly useful for students who are studying Algebra 2 or higher-level math, as it provides a comprehensive look into the graphical representation of trigonometric equations. Understanding these graphs is crucial for various real-world applications, including engineering, physics, and even economics. By mastering these graphing techniques, you can better interpret and apply trigonometric functions in various scenarios.
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Lesson Settings & Tools
| | 15 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Graphing Trigonometric Functions
Slide of 15
Many real-life problems can be solved by applying trigonometric functions, such as sine, cosine, and tangent. This lesson will define these functions, show how to graph them using their function rules, and explore some of their different real-life applications.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Modeling the Population of Rabbits and Foxes
On the weekend, Kriz and their family headed to the local zoo. Kriz really loves learning about animals, so they were sure to stop at all the cool information boards that teach interesting facts about them.
a What type of functions can be used to model the populations of rabbits and foxes?
b Find the appropriate function that models the population of rabbits r(m) as a function of the time m in months.
c Find the appropriate function that models the population of foxes f(m) as a function of the time m in months.
d Graph both functions. One function seems to
chasethe other. What can be a possible explanation for this?
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Trigonometric Functions
Trigonometric functions are functions that relate an input, which represents an acute angle of a right triangle, to a trigonometric ratio of two of the triangle's side lengths. The angle is usually measured in radians.
Trigonometric functions are also defined for angles θ that are not acute by using the appropriate reference angle θ'.
For input values that are not between 0 and 2π, the value of the coterminal angle that belongs to this interval is used instead. Therefore, trigonometric functions are periodic functions.
Because of their close relation with the unit circle, trigonometric functions are also called circular functions. The domain of the sine and cosine functions is the set of all real numbers. Their range is the interval that goes from -1 to 1.
| Trigonometric Function | Domain | Range |
|---|---|---|
| y=sinx | All real numbers | [-1,1] |
| y=cosx | All real numbers | [-1,1] |
The tangent, cotangent, secant, and cosecant functions are defined as rational functions that involve the sine and cosine functions. The domain of each function does not include values that would make their denominator zero.
| Trigonometric Function | Ratio | Domain | Range |
|---|---|---|---|
| y=tanx | tanx=sinx/cosx | Real numbers except odd multiples of π2 | All real numbers |
| y=cotx | cotx=cosx/sinx | Real numbers except multiples of π | All real numbers |
| y=secx | secx=1/cosx | Real numbers except odd multiples of π2 | (- ∞, -1]⋃[1,∞) |
| y=cscx | cscx=1/sinx | Real numbers except multiples of π | (- ∞, -1]⋃[1,∞) |
Now, one of the main trigonometric functions, the sine function, will be defined and examined more closely.
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The Sine Function
Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The sine function, denoted as sin, can be defined as the y-coordinate of the point P.
![The graph of the sine function y=sin(x) over the domain [-2pi,2pi].](/mediawiki/images/0/01/Mljsx_Concept_Sine_Function_2.svg)
Note that for x in the interval [- 2π,0] and in the interval [0,2π] the graph looks exactly the same. This means that the sine function is a periodic function and its period is 2π.
sin(θ+2π n)=sinθ
Here, n is any integer number. Consider the function y=asinbθ, where a and b are non-zero real numbers and θ is measured in radians. With this information, the properties of the sine function can be defined.
| Properties of y=asinbθ | ||
|---|---|---|
| Amplitude | |a| | |
| Number of cycles in [0,2π] | |b| | |
| Period | 2π/|b| | |
| Domain | All real numbers | |
| Range | [- |a|,|a|] | |
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Vertical Displacement of a Buoy
Kriz is interested in a maritime topic and wants to become a sailor one day. Kriz often goes sailing with their father and loves watching how waves crash on the shore, how buoys bob up and down as waves go past, and how the sun slowly melts into the water on the horizon.
Kriz was very surprised when they learned in a math lesson that the vertical displacement of the buoys with respect to the sea level at the nearest beach can be modeled by a trigonometric function.
y=1.6sin π/2t
Here, y is the vertical displacement in feet and t the time in seconds. What are the amplitude, period, and midline of this function?
To calculate the period, substitute π2 for b into the expression 2π|b| and evaluate.
Therefore, the period of the function is 4. This means that every 4 seconds the curve repeats itself, which can be seen on the graph. 
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Hint
In the general form y=asin bθ, the amplitude is |a| and the period is 2π|b|.
Solution
Start by recalling the general form of the sine function. y=asin bθ This function has the following properties.
| Properties of y=asinbθ | ||
|---|---|---|
| Amplitude | |a| | |
| Number of cycles in [0,2π] | |b| | |
| Period | 2π/|b| | |
Now, analyze the given function and identify the values of the coefficients a and b. y= 1.6sin π/2t ⇓ a= 1.6 and b= π/2 Since the value of a is 1.6, the amplitude of the function is |1.6|=1.6 feet. Recall that the midline of the parent sine function is the horizontal line y=0. Since the given function has not been translated up or down, its midline is also y=0.
2π/|b|
Substitute
b= π/2
2π/| π2|
AbsPos
|π/2|=π/2
2π/π2
DivByFracD
a/b/c= a * c/b
4π/π
ReduceFrac
a/b=.a /π./.b /π.
4/1
DivByOne
a/1=a
4
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The Cosine Function
Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The cosine function, denoted as cos, can be defined as the x-coordinate of the point P.
Note that for x in the interval [- 2π,0] and in the interval [0,2π] the graph looks exactly the same. This means that the cosine function is a periodic function and its period is 2π.
cos(θ+2π n)=cosθ
Here, n is any integer number. Consider the function y=acosbθ, where a and b are non-zero real numbers and θ is measured in radians. With this information, the properties of the cosine function can be defined.
| Properties of y=acosbθ | ||
|---|---|---|
| Amplitude | |a| | |
| Number of cycles in [0,2π] | |b| | |
| Period | 2π/|b| | |
| Domain | All real numbers | |
| Range | [- |a|,|a| ] | |
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Operation of Submarines' Radars
Kriz visited the port on a day when a special exhibition was taking place where scientists explained how they use a submarine in ocean exploration. They learned that radars are used to monitor objects under the sea. Even more interesting, in operating radars, sine and cosine functions are involved.
A wave signal received by a radar can be modeled by the following equation. y=3.5cos 12t Here, y is the vertical displacement from the shooting point in centimeters and t is time in seconds. What are the amplitude, period, and midline of this function? Write the period in exact form.
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Hint
In the general form y=acos bθ, the amplitude is |a| and the period is 2π|b|.
Solution
First, recall the general form of a cosine function. y=acos bx This function has the following properties.
| Properties of y=acosbθ | ||
|---|---|---|
| Amplitude | |a| | |
| Number of cycles in [0,2π] | |b| | |
| Period | 2π/|b| | |
Next, examine the given function and identify the values of its coefficients a and b. y= 3.5sin 12t ⇓ a= 3.5 and b= 12 The value of a is 3.5, which means that the amplitude of the function is |3.5|=3.5 centimeters. Recall that the midline of the parent cosine function is y=0. The given function has not been translated vertically, so its midline is also y=0. These two pieces of information can be shown on a graph.
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Identifying Amplitude and Period of Sine and Cosine Functions
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Graphing the Sine and Cosine Functions
Sine and cosine functions can be graphed by closely analyzing their function rules and the graphs of their parent functions, which are y = sin x and y = cos x. As an example, consider the following function.
f(x) = 0.5 cos 2x + 1
In order to graph it, there are four steps to follow.
1
Find the Amplitude, Period, and Translation of the Function
expand_more First, recall the general form of the cosine function and identify the value of each coefficient by comparing it with the given function. y&= acos bθ f(x)& = 0.5 cos 2x + 1 The amplitude of the cosine function is |a| and the period is 2π|b|. Therefore, the amplitude of the given function is | 0.5|=0.5. To find its period, substitute 2 for b into the corresponding expression. 2π/| 2| = π The period of the function is π. Finally, if y=0.5cos 2x is considered, 1 must be added to obtain the given function. f(x)=0.5cos 2x + 1 This means that the function is translated 1 unit upward.
2
Draw the Midline
expand_more The midline of the parent cosine function is y=0. However, since the considered function is translated 1 unit upward, its midline is also translated. This means that the equation of the midline is y=1.
3
Plot Some Key Points on the Graph
expand_more Key points ranging over at least one cycle of the function should now be plotted. These key points are the maximums, minimums, and intersections with the midline. The maximums of y = cos x occur at even multiples of π. x = 0, 2π, 4π, ... The period of f is π, which is 12 of the parent function's period 2π. Also, f has not been translated horizontally, so the maximums neither shifted to right nor to the left. Therefore, the maximums of f occur at the following x-coordinates. x = 12( 0), 12( 2π), 12( 4π), ... ⇕ x = 0, π, 2π, ... This means that the maximums of f occur at multiples of π. Since the midline is y = 1 and the amplitude is 0.5, these maximums will all have a y-value of 1 + 0.5 = 1.5. Now, plot the points on the graph with the midline.
The minimums of f are horizontally located between the maximums. x = 0.5π, 1.5π, 2.5π, ... These points have y-values of 1 - 0.5 = 0.5.
Lastly, in-between every neighboring maximum and minimum are the intersections with the midline. x = 0.25π, 0.75π, 1.25π, ... Since these points lie on the midline, their y-coordinate is 1.
4
Draw the Graph
expand_more By connecting the plotted points with a smooth curve and continuing it periodically in both directions, the graph of the function can finally be drawn.
Extra
Formulas for the Key PointsThere are formulas for the key points such as x -intercepts, maximum value, and minimum value of a sine function of the form y= a sin bx.
| Formula | ||
|---|---|---|
| x-intercepts | (0,0), (1/2*2π/b,0), (2π/b,0) | |
| Maximum | a>0 (1/4 * 2 π/b, a) |
a<0 (3/4*2π/b, - a) |
| Minimum |
a>0 |
a<0 (1/4 * 2 π/b, a) |
Similarly, there are also formulas for the x-intercepts, maximum, and minimum of a cosine function of the form y= a cos bx.
| Formula | ||
|---|---|---|
| x-intercepts | (1/4*2π/b,0), (3/4*2π/b,0) | |
| Maximum | a>0 (0, a), (2π/b, a ) |
a<0 (1/2*2π/b, - a) |
| Minimum | a>0 (1/2*2π/b, - a) |
a<0 (0, a), (2π/b, a ) |
These formulas can be useful when graphing a sine or a cosine function. By using them, the first five points of a function can be plotted. Then, the function can be extended along the x-axis by imitating the found pattern.
Extra
Graphing Parent Sine and Cosine FunctionsThe graph of the parent sine function can be obtained by using a unit circle. Recall that the sine values are represented by the y-coordinate of a point on this circle. Therefore, as the point is rotated, its y-coordinates will be plotted on a coordinate plane.
The graph of the parent cosine function can be drawn in a similar manner. The values of cosine are represented by the x-coordinates of a point on a unit circle. Rotate the point and plot its x-coordinates with the respective θ-values on a coordinate plane.
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Graphing Light Waves of Different Colors
After learning how trigonometric functions are abundant in objects related to the ocean, Kriz was stoked to go to Physics class first thing Monday. There, they learned that light travels in waves and, therefore, can be modeled by sine and cosine functions. Different colors have different wavelengths, or periods, and the amplitude of the wave affects the brightness of the color.
For example, the light visible as red has the longest period, while the light visible as violet has the shortest period. Additionally, the greater the amplitude of the light wave, the brighter it looks.
a Use a sine function to graph the dimmed red light wave with a period of 660 nanometers, an amplitude of 0.3 units, and whose midline is y=0.5.
b Graph the bright violet light wave modeled by the following equation.
y=1.4cos π/200x
Answer
a
b
Hint
a First, plot the midline and then identify the locations of the maximums, minimums, and the intersections with the midline.
b Identify the values of a and b and use the fact the period is 2π|b|. Analyze the locations of maximums, minimums, and intersections with the midline of the parent function of cosine.
Solution
a The first step is to graph the midline of the function, which is said to be y=0.5.
Next, some key points, like maximums, minimums, and intersections with the midline should be plotted. The parent sine function intersects the midline at each half-period.
In this case, the period is 660, so its half-period is 6602=330 nanometers. Therefore, the x-coordinates of the intersections of the function and the midline occur at x-values that are multiples of 330. x=0, 330, 660, ... These points lie on the midline, so their y-coordinate is 0.5.
The maximums and minimums of a sine function occur once every period between two points of intersection with the midline. Analyzing the graph of the parent sine function starting from the origin, it can be seen that the maximum of the function occurs before the minimum.
Therefore, the maximum of the given function is in the middle between the intersections (0,0.5) and (330,0.5), while the minimum is in the middle between (330,0.5) and (660,0.5). By adding and subtracting the amplitude of 0.3 to the midline, the y-coordinates of the maximum and minimum, respectively, can be found. cc Maximum& Minimum [0.2cm] (330/2,0.5 + 0.3) & (330 + 330/2,0.5 - 0.3) ⇕ & ⇕ (165,0.8) & (495,0.2) Now, plot both points on the coordinate plane.
Finally, connect the points with a smooth curve and continue it periodically.
b To graph the given function, start by comparing it with the general form of a cosine function to identify the values of the coefficients.
Period=2π/|b|
Substitute
b= π/200
Period=2π/| π200|
▼
Evaluate right-hand side
AbsPos
|π/200|=π/200
Period=2π/π200
DivByFracD
a/b/c= a * c/b
Period=400π/π
ReduceFrac
a/b=.a /π./.b /π.
Period=400/1
DivByOne
a/1=a
Period=400
Next, the key points should be identified and plotted. Consider the parent cosine function.
As can be seen, the maximums occur at x=- 2π, 0, 2π,..., which are the multiples of its period 2π. This means that using the period of 400 of the considered function, its maximums can be found. Maximums x=0, 400, 800, ... y=1.4 Furthermore, since the equation of the midline is y=0 and the amplitude is 1.4, the y-coordinate of the maximums is 1.4. The minimums of the parent function occur at x=- π, π,..., which are the values of its half-period. In this case, the half-period of the function is 200. Minimums x=200, 600, 1000, ... y=- 1.4 The y-coordinates of the minimums are - 1.4. Lastly, in-between every neighboring maximum and minimum are the intersections with the midline. Intersections x=100, 300, 500, ... y=0 Finally, plot all the found points on the coordinate plane with the midline.
By connecting the points with a smooth curve and continuing it periodically, the graph of the given function can be obtained.
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Frequency of a Periodic Function
The frequency of a periodic function is the number of cycles in a given unit of time. The frequency of a function's graph is the reciprocal of the function's period.
Frequency=1/Period
hertz.For instance, 10 Hz means 10 times per second.
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Frequencies That Animals Can Hear
Later that day, Kriz was excitingly sharing their impressions with their classmate Zain about their visit to the zoo. Kriz told Zain that they were impressed to learn that elephants can hear frequencies 20 times lower than humans, while mice can hear astronomically high frequencies, up to 70 - 80 kHz.
a Write a sine function in the form y=asin bx, with a and b as positive real numbers, that models a sound wave with a frequency of 10 Hz and an amplitude of 1 unit that elephants can hear.
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b Write a cosine function in the form y=acos bx, with a and b positive real numbers, that models the sound wave with a frequency of 75 kHz and an amplitude of 0.2 units that mice can hear.
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Solution
a Start by recalling the general form of a sine function.
Frequency=1/Period
Substitute
Frequency= 10
10=1/Period
MultEqn
LHS * Period=RHS* Period
10* Period=1
DivEqn
.LHS /10.=.RHS /10.
Period=1/10
Period=2π/b
Substitute
Period= 1/10
1/10=2π/b
CrossMult
Cross multiply
1* b = 10* 2π
Multiply
Multiply
b=20π
hearwhen communicating miles apart!
b Similarly to Part A, first recall the general form of a cosine function.
Frequency=1/Period
Substitute
Frequency= 75 000
75 000=1/Period
MultEqn
LHS * Period=RHS* Period
75 000* Period=1
DivEqn
.LHS /75 000.=.RHS /75 000.
Period=1/75 000
Period=2π/b
Substitute
Period= 1/75 000
1/75 000=2π/b
CrossMult
Cross multiply
1* b = 75 000* 2π
Multiply
Multiply
b=150 000π
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Investigating the Graphs of the Functions Involving Sine and Cosine
It is interesting to explore the graphs of functions that are defined by applying some basic operations, like addition, multiplication, and division, to sine and cosine functions. First, try to draw the graph of f(x)=sin x+cos x.
Now, the graph of f(x)=sin xcos x will be drawn.
Finally, by applying the same method, try to draw the graph of f(x)= sin xcos x. 
What function does the obtained graph resemble?
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The Tangent Function
Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The tangent function, denoted as tan, can be defined as the ratio of the y-coordinate to the x-coordinate of the point P.
tanθ=sinθ/cos θ
The graph of the tangent function is as follows.
![The graph of the tangent function y=tan(x) with asymptotes at -3pi/2, -pi/2, pi/2, 3pi/2, and 5pi/2 over the domain [-3pi/2,5pi/2]](/mediawiki/images/8/80/Mljsx_Concept_Tangent_Function_2.svg)
The period of the tangent function is π. Since each branch comes from negative infinity towards positive infinity, the tangent function has no amplitude and its range is all real numbers. Consider the function y=atanbx, where a and b are non-zero real numbers and θ is measured in radians. The properties of the tangent function can be identified from the function rule.
| Properties of y=atanbx | |
|---|---|
| Amplitude | No amplitude |
| Interval of One Cycle | (- π/2|b|,π/2|b|) |
| Asymptotes | At the end of each cycle |
| Period | π/|b| |
| Domain | All real numbers except odd multiples of π/2|b| |
| Range | All real numbers |
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Graphing a Tangent Function
A tangent function can be graphed by examining its function rule and determining some of its key characteristics, like period and asymptotes. Consider the following function. y=1.5tan π/2x
In order to draw the graph, there are four steps to follow.
Finally, the branches are drawn by starting from the bottom of the left asymptote and moving towards the top of the right asymptote.
1
Find the Period of the Function
expand_more Start by comparing the function with the general form of a tangent function to identify the value of its coefficients.
General Form:& y= atan bx Given Function:& y= 1.5tan π/2x
The period of a tangent function is given by π|b|. Substitute π2 for b and calculate the period of the given function.
Therefore, the period of the function is 2 units.
π/|b|
Substitute
b= π/2
π/| π2|
AbsPos
|π/2|=π/2
π/π2
DivByFracD
a/b/c= a * c/b
2π/π
CalcQuot
Calculate quotient
2
2
Draw the Asymptotes
expand_more Recall that one cycle of a tangent function occurs in the interval from - π2|b| to π2|b|. Again, substitute the value of b and find the interval corresponding to the considered function.
One cycle of the function occurs in the interval [- 1,1]. Furthermore, recall that the asymptotes occur at the end of each cycle of the function. Therefore, the function has asymptotes at x=- 1 and x=1, which can be drawn on a coordinate plane. 
π/2|b|
Substitute
b= π/2
π/2| π2|
AbsPos
|π/2|=π/2
π/2( π2)
DenomMultFracToNumber
2 * a/2= a
π/π
QuotOne
a/a=1
1
3
Plot Some Points on the Graph
expand_more Next, divide the period of the function into four equal parts and locate three points between the asymptotes. In this case, the period is 2, so each fourth is 0.5 units long.
Therefore, the x-coordinates of the points that will be plotted are - 12, 0, and 12. Substitute these values for x into the function rule and evaluate the corresponding y-coordinates.
| y=1.5tan π2x | |||
|---|---|---|---|
| x-Coordinate | Substitute | Simplify | Evaluate |
| x= - 1/2 | y=1.5tan π/2( - 1/2) | y=1.5tan (- π/4) | y=- 1.5 |
| x= 0 | y=1.5tan π/2( 0) | y=1.5tan 0 | y=0 |
| x= - 1/2 | y=1.5tan π/2( 1/2) | y=1.5tan π/4 | y=1.5 |
Now that both coordinates of the three points are known, plot them on the coordinate plane with the asymptotes.
4
Draw the Graph
expand_more Finally, connect the three points with a smooth curve and continue the function to the left and right keeping in mind that it should get closer to the asymptotes but will never intersect them.
Replicate the branch to obtain the graph for other intervals.
Extra
Key Characteristics of a Tangent FunctionWhen drawing one period of a function of the form y= a tan bθ, the following characteristics of a tangent function can be used.
| Formula | ||
|---|---|---|
| x-intercept | (0,0) | |
| Asymptotes | θ = - π/2| b|, θ = π/2| b| | |
| Halfway Points | (- π/4 b,- a), (π/4 b, a) | |
Extra
Graphing the Parent Tangent FunctionRecall that a tangent function can be defined as the quotient of a sine and a cosine function. Therefore, the graph of the parent tangent function can be drawn as a rational function. tan x = sin x/cos x First, draw the graphs of the sine and cosine functions.
Because the tangent function has cosine in its denominator, the asymptotes of the tangent function are located at the zeros of the cosine function.
Since the sine function is the numerator of the tangent function, the zeros of the sine function will be the zeros of the tangent function as well.
The y-values of the sine and cosine functions are equal at the intersection of both graphs. This means that the y-value of the tangent function will be 1 at the x-coordinates of these points.
Likewise, three more points can be plotted between the left asymptotes of each period and the zeros. This time, however, since the sine and cosine functions have opposite x-values, the y-value of the tangent function will be - 1.
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Modeling a Distance with a Tangent Function
After school, Kriz and Zain will meet some friends to play volleyball near Zain's home. Zain lives in a modern 300-feet building. As Kriz was approaching the building about 175 feet from its base, they saw Zain going down in the elevator and waved to them.
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a Write an equation that models Zain's distance d in feet from the top of the building as a function of the angle of elevation x.
b Graph one cycle of the function from Part A ignoring the limitations of the context.
c What part of the graph makes sense considering the given context?
Answer
a d=300-175tan x
b
c The part in Quadrant I.
Hint
a Which trigonometric function relates the legs of a right triangle with an acute angle? Set up an equation and solve it for d.
b Use the fact that the period of a tangent function is given by π|b| and one cycle occurs in the interval from - π2|b| to π2|b|.
c Recall what x and d represent. What values cannot they have?
Solution
a Begin by analyzing the triangle formed by Kriz, Zain and the base of the building.
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tan x=300-d/175
MultEqn
LHS * 175=RHS* 175
175tan x=300-d
AddEqn
LHS+d=RHS+d
d+175tan x=300
SubEqn
LHS-175tan x=RHS-175tan x
d=300-175tan x
b The function found in Part A can be seen as a vertical translation of the function y=- 175 tan x by 300 units up.
d= 300-175tan x Therefore, to obtain the graph of d, first graph y=- 175 tan x and then translate it vertically. Start by comparing the function to the general form of a tangent function to identify the values of the coefficients a and b. General Form:& y= atan bx Given Function:& y= - 175 tan 1x Recall that the period of the parent tangent function is given by π|b|. Since b= 1, the period of the function can be calculated. Period π/| b| ⇒ π/| 1|=π One cycle of a tangent function occurs in the interval from - π2|b| to π2|b|. Substitute the value of b and find this interval for the given function. (- π/2| 1|,π/2| 1|) ⇔ (- π/2,π/2) Additionally, the asymptotes occur at the end of each cycle. This means that two asymptotes are x=- π2 and x= π2. Graph them on a coordinate plane.
Next, three points should be plotted between the asymptotes. By dividing the interval between - π2 and π2 into four equal parts, the x-coordinates of three points are obtained. x=- π/4, x=0, x=π/4 Substitute them into the function rule for x and find the corresponding y-coordinates.
| y=- 175tan x | |||
|---|---|---|---|
| x-Coordinate | Substitute | Evaluate | |
| x= - π/4 | y=- 175tan ( - π/4) | y=175 | |
| x= 0 | y=- 175tan 0 | y=0 | |
| x= π/4 | y=- 175tan ( π/4) | y=- 175 | |
Now, plot the points on the coordinate plane with the asymptotes.
Finally, the graph of y can be drawn by connecting the points with a smooth curve such that, as x tends to - π2 and π2, the graph gets closer and closer to the asymptotes but never crosses them.
The last step is to translate the graph 300 units up vertically to obtain the graph of d.
c To determine which part of the graph makes sense in the context of the given situation, recall that d represents the distance between Zain and the top of the building, while x is the measure of the angle of elevation. In real life, the values of these two variables can only be greater than or equal to 0.
Therefore, only the part of the graph in the first quadrant makes sense considering the context.
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Drawing Conclusions About the Population of Rabbits and Foxes
Earlier, it was mentioned that on the weekend Kriz went with their family to the local zoo.
External credits: @storyset
a What type of functions can be used to model the populations of rabbits and foxes?
b Find the appropriate function to model the population of rabbits r(m) as a function of the time m in months.
c Find the appropriate function to model the population of foxes f(m) as a function of the time m in months.
d Graph both functions. One function seems to chase the other. What can be a possible explanation for this?
Answer
a Sine or cosine functions.
b r(m)=- 500sin π/6m+1250
c f(m)=50cos π/6m+200
d Graph of r(m):
Graph of f(m):
Explanation: A predator-prey relationship between foxes and rabbits.
Hint
a First, analyze the values in the table and try to find patterns. Then, plot the points and examine the shape of the curve that connects them.
c The equation of the midline indicates the number of units by which the function has to be translated vertically.
d Connect the given and plotted points with a smooth curve. Analyze what happens to the population of foxes when the population of rabbits increases and decreases, and vice versa.
Solution
a Begin by analyzing the values of the rabbit population given in the table. Are there any patterns?
Both functions have a general shape of a sine or a cosine function.
b Start by examining the shape of the graph in order to determine some of its characteristics, like midline, period, and amplitude.
On the graph almost one full cycle of the function is shown and it can be assumed that it will repeat periodically moving to the right along the x-axis. By analyzing the graph's pattern, it can be concluded that for x=12, the function will have a value of 1250, the same value as for x=0. Therefore, the function has a period of 12 months. Period 12 months Additionally, the maximum value is 1750, while the minimum is 750. By calculating the mean of these values, the amplitude can be found. Amplitude [0.2cm] 1750- 750/2=500 Next, by subtracting the value of the amplitude from the maximum, or by adding its value to the minimum, the equation of the midline can be determined. Midline y=1250 Show all of these characteristics on the graph and try to identify which function, sine or cosine, can represent it.
Period=2π/|b|
▼
Substitute 12 for Period and evaluate
Substitute
Period= 12
12=2π/|b|
MultEqn
LHS * |b|=RHS* |b|
12|b|=2π
DivEqn
.LHS /12.=.RHS /12.
|b|=2π/12
ReduceFrac
a/b=.a /2./.b /2.
|b|=π/6
c Similar to Part B, the function that models the population of foxes can be written. Again, begin by analyzing the shape of the graph and try to draw some conclusions about it.
Since almost one full cycle of the function is shown, it can be assumed that it will repeat periodically moving to the right along the x-axis. Also, it can be concluded that for x=12, the function will have the value 250, the same value as for x=0. This means that the function has a period of 12 months. Period 12 months The maximum value is y= 250, while the minimum is 150. Calculate the mean of these values to find the amplitude of the function. Amplitude [0.2cm] 250- 150/2=50 Now, by subtracting the value of the amplitude from the maximum, or by adding its value to the minimum, the equation of the midline can be written. Midline y=200 Show all of these characteristics on the graph and try to identify which function, sine or cosine, can represent it.
The diagram reminds the graph of a cosine function. Recall the general form of a cosine function. y=acos bm Here, a represents the amplitude and b represents the coefficient related to the period by the following formula. Period=2π/|b| Since for this function period is also 12 months, the value of |b| is π6. Taking into account only the positive values of a and b, substitute them into the equation. y= acos bm ⇓ y= 50cos π/6m Furthermore, the midline at y=200 indicates that the function has been translated 200 units up. This fact can be shown in the function rule by adding 200 to the right-hand side. f(m)=50cos π/6m+ 200 The function that models the population of foxes during the previous year was obtained using the graph and the values from the table, making it possible to determine the needed characteristics.
d To graph r(m), connect the given points plotted in Part A with a smooth curve.
Similarly, by connecting the points given for the function f(m), its graph can be drawn.
The fox population seems to be behind the rabbit population by three months. In other words, the fox population chases the rabbit population. This can be explained by the predator-prey relationship that exists between foxes and rabbits, as rabbits are a source of food for foxes. Where there is a big population of rabbits, the fox population grows. Many rabbits → Foxes ↑ At the same time, when there are a lot of foxes, more rabbits are being hunted, so the rabbit population decreases. Many foxes → Rabbits ↓ When the rabbit population gets quite small, foxes have a limited food supply and, therefore, their population decreases. Few rabbits → Foxes ↓ Once the fox population is small enough, the rabbit population can recover, and this cycle continues. Few foxes → Rabbits ↑
Graphing Trigonometric Functions
Exercises
Consider the following trigonometric function. y = 7.1 sin 11 t
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To find the amplitude of the given function, let's recall the general form of the sine function. y=asin bx This function has the following properties.
| Properties of y=asinbθ | ||
|---|---|---|
| Amplitude | |a| | |
| Number of cycles in [0,2π] | |b| | |
| Period | 2π/|b| | |
Now we will examine the given function to identify the values of its coefficients a and b. y = 7.1 sin 11 t ⇓ a= 7.1 and b= 11 We can see that the value of a is 7.1, which means that the function has an amplitude of |7.1|=7.1.
Recall that the midline of the parent sine function changes when the function is vertically translated. Let's recall how a vertical translation of the sine function looks.
y=asin bx + h
The constant h above indicates that the function is vertically translated h units. Now let's compare it to our given function.
y = 7.1 sin 11t + 0
Looking at the given function, we can see that it has not been translated vertically, so its midline is the same as the parent sine function, y=0.
To calculate the period, we need to substitute 11 for b into the expression 2π|b| and simplify. Let's do it!
The period of the function is 2π11. This means that the graph of the function repeats itself every 2π11 units.
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Use a reference angle to find the exact value of the following trigonometric function.
sin 585^(∘)
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We want to find the exact value of sin 585^(∘). To do so, let's start by recalling some trigonometric values for special angles.
| Trigonometric Values for Special Angles | |||||
|---|---|---|---|---|---|
| Sine | Cosine | Tangent | Cosecant | Secant | Cotangent |
| sin 30^(∘)=1/2 | cos 30^(∘)=sqrt(3)/2 | tan 30^(∘)=sqrt(3)/3 | csc 30^(∘)=2 | sec 30^(∘)=2sqrt(3)/3 | cot 30^(∘)=sqrt(3) |
| sin 45^(∘)=sqrt(2)/2 | cos 45^(∘)=sqrt(2)/2 | tan 45^(∘)=1 | csc 45^(∘)=sqrt(2) | sec 45^(∘)=sqrt(2) | cot 45^(∘)=1 |
| sin 60^(∘)=sqrt(3)/2 | cos 60^(∘)=1/2 | tan 60^(∘)=sqrt(3) | csc 60^(∘)=2sqrt(3)/3 | sec 60^(∘)=2 | cot 60^(∘)=sqrt(3)/3 |
As we can see, the value we need is not on this table. To find the value, let's graph θ =585^(∘) in standard position so that we can find its reference angle. Be aware that the measure of our angle is greater than 360^(∘), so the angle is greater than a full turn.
If the measure of θ is greater than 360^(∘) or less than 0^(∘), we use a coterminal angle with a positive measure between 0^(∘) and 360^(∘) to find the reference angle. To do so, we will subtract 360^(∘) from the measure of the given angle.
Now we can find the reference angle θ ' of θ =585^(∘). This way we can use the values from our table. Note that the terminal side of 225^(∘) lies in Quadrant III. Therefore, to find its reference angle θ', we will subtract 180^(∘) from 225^(∘).
We found that θ '=45^(∘). Next, we will recall the signs of the six trigonometric functions in the different quadrants of the coordinate plane.
In Quadrant III — the quadrant where the terminal side of the angle is located — the sine is negative. With this information, we can write an equation relating the sine of the angle and the sine of its reference angle. sin 585^(∘) = - sin 45^(∘) Using our table, we can see that sin 45^(∘)= sqrt(2)2. sin 585^(∘) = - sin 45^(∘) ⇒ sin 585^(∘) =- sqrt(2)/2
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The following point belongs to the unit circle. P(sqrt(2)/4,- sqrt(14)/4) The terminal side of an angle θ in standard position passes through P.
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A unit circle is a circle of radius 1 centered at the origin on the coordinate plane. The cosine of an angle in standard position whose terminal side intersects the unit circle at point P is the first coordinate of P.
We are told that the terminal side of angle θ in standard position goes through P and that P lies on the unit circle. Therefore, the terminal side of the angle intersects the unit circle at P( sqrt(2)4, -sqrt(14)4). Using this information, we can identify the cosine of θ. Let's do it! cos θ= sqrt(2)/4
We will proceed in the same fashon as we did in Part A. However, the sine of an angle in standard position whose terminal side intersects the unit circle at point P is the second coordinate of P. Let's identify the sine of θ! sin θ= -sqrt(14)/4
Let's draw a graph to represent the given situation. Be aware that P is located in Quadrant IV.
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Consider the following graph of a sine function y(x).
The values of x are angles in radians.
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We want to use the graph of the sine function to find the value of y(x) for x =π radians. Let's find the point on the curve with an x-coordinate of π and look for its corresponding y-coordinate.
We can see that the value of y(x) for x = π radians is 2.
This time we will look at the graph of y(x) to find the value of y(x) for x =-π radians. Let's find the point on the curve with an x-coordinate of -π and look for its corresponding y-coordinate.
We can see that the value of y(x) for x = -π radians is also 2.
Finally, we will examine the graph of y(x) to find the value of y(x) for x = 3π2 radians. Let's find the point on the curve with an x-coordinate of 3π2 and look for its corresponding y-coordinate.
We can see that the value of y(x) for x = 3π2 radians is 0. We did it!
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Consider the following graph of a sine function.
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We want to use the given graph to determine the number of cycles that the sine function has in the interval from 0 to 2π. To determine the number of cycles, we need to identify when the shape of the graph starts repeating. Let's do it!
Looking at the graph, we can see that the graph repeats two times for the values between 0 and 2π. Therefore, there are two cycles in this interval.
Now we need to find the amplitude of the function. Let's review the definition of this term before we look at the graph.
Amplitude |- The amplitude is half the difference of the maximum and minimum values of a periodic function.
Let's identify the minimum and the maximum of the function in the graph!
We can see that the maximum and minimum values are 3 and - 3, respectively. Now we can find the amplitude!
The period is the horizontal length of one cycle. In Part A we found that there are two cycles in the interval from 0 to 2π. Therefore, we can find the period by dividing 2π by 2. Let's do it! Period = 2π/2 ⇒ Period = π
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Graphing Trigonometric Functions
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