# 6.1 Graphs of the sine and cosine functions  (Page 7/13)

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## Determining a rider’s height on a ferris wheel

The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes.

With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.

Passengers board 2 m above ground level, so the center of the wheel must be located $\text{\hspace{0.17em}}67.5+2=69.5\text{\hspace{0.17em}}$ m above ground level. The midline of the oscillation will be at 69.5 m.

The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.

Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.

• Amplitude: $\text{\hspace{0.17em}}\text{67}\text{.5,}\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}A=67.5$
• Midline: $\text{\hspace{0.17em}}\text{69}\text{.5,}\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}D=69.5$
• Period: $\text{\hspace{0.17em}}\text{30,}\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}B=\frac{2\pi }{30}=\frac{\pi }{15}$
• Shape: $\text{\hspace{0.17em}}\mathrm{-cos}\left(t\right)$

An equation for the rider’s height would be

$y=-67.5\mathrm{cos}\left(\frac{\pi }{15}t\right)+69.5$

where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in minutes and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is measured in meters.

Access these online resources for additional instruction and practice with graphs of sine and cosine functions.

## Key equations

 Sinusoidal functions $\begin{array}{l}f\left(x\right)=A\mathrm{sin}\left(Bx-C\right)+D\\ f\left(x\right)=A\mathrm{cos}\left(Bx-C\right)+D\end{array}$

## Key concepts

• Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of $\text{\hspace{0.17em}}2\pi .$
• The function $\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is odd, so its graph is symmetric about the origin. The function $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is even, so its graph is symmetric about the y -axis.
• The graph of a sinusoidal function has the same general shape as a sine or cosine function.
• In the general formula for a sinusoidal function, the period is $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}.\text{\hspace{0.17em}}$ See [link] .
• In the general formula for a sinusoidal function, $\text{\hspace{0.17em}}|A|\text{\hspace{0.17em}}$ represents amplitude. If $\text{\hspace{0.17em}}|A|>1,\text{\hspace{0.17em}}$ the function is stretched, whereas if $\text{\hspace{0.17em}}|A|<1,\text{\hspace{0.17em}}$ the function is compressed. See [link] .
• The value $\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ in the general formula for a sinusoidal function indicates the phase shift. See [link] .
• The value $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ in the general formula for a sinusoidal function indicates the vertical shift from the midline. See [link] .
• Combinations of variations of sinusoidal functions can be detected from an equation. See [link] .
• The equation for a sinusoidal function can be determined from a graph. See [link] and [link] .
• A function can be graphed by identifying its amplitude and period. See [link] and [link] .
• A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See [link] .
• Sinusoidal functions can be used to solve real-world problems. See [link] , [link] , and [link] .

## Verbal

Why are the sine and cosine functions called periodic functions?

The sine and cosine functions have the property that $\text{\hspace{0.17em}}f\left(x+P\right)=f\left(x\right)\text{\hspace{0.17em}}$ for a certain $\text{\hspace{0.17em}}P.\text{\hspace{0.17em}}$ This means that the function values repeat for every $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ units on the x -axis.

How does the graph of $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ compare with the graph of $\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x?\text{\hspace{0.17em}}$ Explain how you could horizontally translate the graph of $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to obtain $\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x.$

the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
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
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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
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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?
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This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
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Jeffrey
Confunction Identity
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meena
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard