<< Chapter < Page Chapter >> Page >

Using the graphs of trigonometric functions to solve real-world problems

Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function .

Using trigonometric functions to solve real-world scenarios

Suppose the function y = 5 tan ( π 4 t ) marks the distance in the movement of a light beam from the top of a police car across a wall where t is the time in seconds and y is the distance in feet from a point on the wall directly across from the police car.

  1. Find and interpret the stretching factor and period.
  2. Graph on the interval [ 0 , 5 ] .
  3. Evaluate f ( 1 ) and discuss the function’s value at that input.
  1. We know from the general form of y = A tan ( B t ) that | A | is the stretching factor and π B is the period.
    A graph showing that variable A is the coefficient of the tangent function and variable B is the coefficient of x, which is within that tangent function.

    We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.

    The period is π π 4 = π 1 4 π = 4. This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.

  2. To graph the function, we draw an asymptote at t = 2 and use the stretching factor and period. See [link]
    A graph of one period of a modified tangent function, with a vertical asymptote at x=4.
  3. period: f ( 1 ) = 5 tan ( π 4 ( 1 ) ) = 5 ( 1 ) = 5 ; after 1 second, the beam of has moved 5 ft from the spot across from the police car.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with graphs of other trigonometric functions.

Key equations

Shifted, compressed, and/or stretched tangent function y = A tan ( B x C ) + D
Shifted, compressed, and/or stretched secant function y = A sec ( B x C ) + D
Shifted, compressed, and/or stretched cosecant function y = A csc ( B x C ) + D
Shifted, compressed, and/or stretched cotangent function y = A cot ( B x C ) + D

Key concepts

  • The tangent function has period π .
  • f ( x ) = A tan ( B x C ) + D is a tangent with vertical and/or horizontal stretch/compression and shift. See [link] , [link] , and [link] .
  • The secant and cosecant are both periodic functions with a period of 2 π . f ( x ) = A sec ( B x C ) + D gives a shifted, compressed, and/or stretched secant function graph. See [link] and [link] .
  • f ( x ) = A csc ( B x C ) + D gives a shifted, compressed, and/or stretched cosecant function graph. See [link] and [link] .
  • The cotangent function has period π and vertical asymptotes at 0 , ± π , ± 2 π , ... .
  • The range of cotangent is ( , ) , and the function is decreasing at each point in its range.
  • The cotangent is zero at ± π 2 , ± 3 π 2 , ... .
  • f ( x ) = A cot ( B x C ) + D is a cotangent with vertical and/or horizontal stretch/compression and shift. See [link] and [link] .
  • Real-world scenarios can be solved using graphs of trigonometric functions. See [link] .

Section exercises

Verbal

Explain how the graph of the sine function can be used to graph y = csc x .

Since y = csc x is the reciprocal function of y = sin x , you can plot the reciprocal of the coordinates on the graph of y = sin x to obtain the y -coordinates of y = csc x . The x -intercepts of the graph y = sin x are the vertical asymptotes for the graph of y = csc x .

Got questions? Get instant answers now!

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask