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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.
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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 .

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Questions & Answers

difference between calculus and pre calculus?
Asma Reply
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x³+y³+z³=42
Robert
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Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
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Quera Reply
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Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
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can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
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Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
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Loree
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jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
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jeric Reply
What are the question marks for?
Elliott

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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