# 8.1 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 $\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x.$

#### Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
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hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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find the sum of 28th term of the AP 3+10+17+---------
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I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
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write down the polynomial function with root 1/3,2,-3 with solution
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if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
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In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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the polar co-ordinate of the point (-1, -1)
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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