# 6.2 Graphs of the other trigonometric functions  (Page 8/9)

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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 $\text{\hspace{0.17em}}y=5\mathrm{tan}\left(\frac{\pi }{4}t\right)\text{\hspace{0.17em}}$ marks the distance in the movement of a light beam from the top of a police car across a wall where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the time in seconds and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\left[0,5\right].$
3. Evaluate $\text{\hspace{0.17em}}f\left(1\right)\text{\hspace{0.17em}}$ and discuss the function’s value at that input.
1. We know from the general form of $\text{\hspace{0.17em}}y=A\mathrm{tan}\left(Bt\right)\text{\hspace{0.17em}}$ that $\text{\hspace{0.17em}}|A|\text{\hspace{0.17em}}$ is the stretching factor and $\text{\hspace{0.17em}}\frac{\pi }{B}\text{\hspace{0.17em}}$ is the period.

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 $\text{\hspace{0.17em}}\frac{\pi }{\frac{\pi }{4}}=\frac{\pi }{1}\cdot \frac{4}{\pi }=4.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}t=2\text{\hspace{0.17em}}$ and use the stretching factor and period. See [link]
3. period: $\text{\hspace{0.17em}}f\left(1\right)=5\mathrm{tan}\left(\frac{\pi }{4}\left(1\right)\right)=5\left(1\right)=5;\text{\hspace{0.17em}}$ after 1 second, the beam of has moved 5 ft from the spot across from the police car.

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

## Key equations

 Shifted, compressed, and/or stretched tangent function $y=A\text{\hspace{0.17em}}\mathrm{tan}\left(Bx-C\right)+D$ Shifted, compressed, and/or stretched secant function $y=A\text{\hspace{0.17em}}\mathrm{sec}\left(Bx-C\right)+D$ Shifted, compressed, and/or stretched cosecant function $y=A\text{\hspace{0.17em}}\mathrm{csc}\left(Bx-C\right)+D$ Shifted, compressed, and/or stretched cotangent function $y=A\text{\hspace{0.17em}}\mathrm{cot}\left(Bx-C\right)+D$

## Key concepts

• The tangent function has period $\text{\hspace{0.17em}}\pi .$
• $f\left(x\right)=A\mathrm{tan}\left(Bx-C\right)+D\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}2\pi .\text{\hspace{0.17em}}$ $f\left(x\right)=A\mathrm{sec}\left(Bx-C\right)+D\text{\hspace{0.17em}}$ gives a shifted, compressed, and/or stretched secant function graph. See [link] and [link] .
• $f\left(x\right)=A\mathrm{csc}\left(Bx-C\right)+D\text{\hspace{0.17em}}$ gives a shifted, compressed, and/or stretched cosecant function graph. See [link] and [link] .
• The cotangent function has period $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ and vertical asymptotes at $\text{\hspace{0.17em}}0,±\pi ,±2\pi ,....$
• The range of cotangent is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),\text{\hspace{0.17em}}$ and the function is decreasing at each point in its range.
• The cotangent is zero at $\text{\hspace{0.17em}}±\frac{\pi }{2},±\frac{3\pi }{2},....$
• $f\left(x\right)=A\mathrm{cot}\left(Bx-C\right)+D\text{\hspace{0.17em}}$ 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] .

## Verbal

Explain how the graph of the sine function can be used to graph $\text{\hspace{0.17em}}y=\mathrm{csc}\text{\hspace{0.17em}}x.$

Since $\text{\hspace{0.17em}}y=\mathrm{csc}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the reciprocal function of $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ you can plot the reciprocal of the coordinates on the graph of $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to obtain the y -coordinates of $\text{\hspace{0.17em}}y=\mathrm{csc}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ The x -intercepts of the graph $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ are the vertical asymptotes for the graph of $\text{\hspace{0.17em}}y=\mathrm{csc}\text{\hspace{0.17em}}x.$

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?