<< 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

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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