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GIVE YOUR POST A TITLE THAT WILL GIVE READERS AN IDEA OF WHAT YOUR APPLICATIONS ARE ABOUT

(0) Post your name

(1) In 75–125 words, summarize an application of PERIODIC FUNCTIONS you found while searching online.

(2) Insert the link(s) to any articles, videos, podcasts, or other media where you found the application of trigonometry.

(3) Why did you want to learn about this particular application of trigonometry? (Maybe because you foresee yourself using this in the future? Or maybe you found this application simply fascinating? Whatever the case, explain.)

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1. Sound waves are one of the most popular periodic functions in nature. The way in which light travels through or reflects from different surfaces varies but they are all periodic. The wavelength and frequency can change depending on the color but they travel in a wave-like shape, just like a periodic function. They also kind of relate to what we saw on the ferris wheel function, since the amplitude or diameter of the wave can change.
3. Might be able to use this knowledge in the future because (if i were to stick with studying architecture) knowing how to manipulate light and colors is important.

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When a pressure wave reaches the eardrum, the wave causes the eardrum to vibrate at the same frequency as the wave. The inner ear and the brain convert this vibration into a perceived sound. Since sound is a periodic pressure wave, the sine function is often used to model sound mathematically.

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BEATRIZ DIAQUE

1. Measuring the tide involves periodic functions since scientists can measure the height of the water at different times of the day. Trigonometric functions could also be applied in this case to model and predict the high or the low tides. This knowledge could be further expanded in order to apply it to measuring waves during hurricanes and storms, e.g. tsunami.

2. https://flexbooks.ck12.org/cbook/ck-12-interactive-algebra-2-for-ccss/section/4.7/primary/lesson/modeling-periodic-behavior-alg-2-ccss/

3. This particular application of trigonometry was interesting to me because it helped me understand how math is involved in real life. The ocean, hurricanes, storms, etc, are all elements of nature, and I found it very interesting that it could be broken down into a periodic function.

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1) Tide is something that is changes periodically, sometimes more than once in a day. Hence, it changes periodically and can be predicted it can be graphed through a periodic function. Tides are caused by the moons gravitatinal pull and it's position and sence the moon's position can be pridicted then tides can also be predicted.

2)https://www.dummies.com/education/math/trigonometry/measure-tidal-change-using-a-trigonometry-graph/
3) I wanted to research tides specifically as I have been to places where the changes are extremely noticeable, so to learn and now that it can be predicted and graphed is interesting.

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Dummies has always stood for taking on complex concepts and making them easy to understand. Dummies helps everyone be more knowledgeable and confident in applying what they know. Whether it's to pass that big test, qualify for that big promotion or even master that cooking technique; people who rely on dummies, rely on it to learn the critical skills and relevant information necessary for success.

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Measure Tidal Change Using a Trigonometry Graph - dummies

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You can use trigonometry to graph the changes in high and low tides for a particular location. Along the coast, the tides are of particular interest. They are affected by the gravitational pull of both the moon and the sun. The high tides and low tides follow a periodic pattern that you can model with [...]

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1. normal breathing involves rhythmic inhaling and exhaling. therefore, we can count breathing using trigonometric equations. changes in breathing can also be measured. some of the variables used to measure it are lung capacity, time, amplitude, period, phase shift. you can use trigonometry to measure the quality of lungs and their air capacity over time and how it affects the flow of your breathing. It can of course change but then we can see how much and can maybe lead to why it changed and then maybe find a solution
2. http://www.tiem.utk.edu/~gross/bioed/webmodules/lungcapacity.html\
3. I think that this could be another way to measure a person and keep track of their health. I find it similar to a heart rate and as a way to measure. I can also see this used in future surgery environments since it could be another way to check if the patient is doing ok in case the heart rate sensor fails or malfunctions.

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RHYTHMS IN BREATHING AND LUNG CAPACITY

Introduction: The respiratory system functions to filter, warm, and humidify the air we breathe, and to supply cells with oxygen while removing carbon dioxide. In animals that breathe with lungs, air moves into the lungs through the trachea and then back out again. When each breath is completed, the lung still contains a volume of air, called the residual volume. Each inhalation adds additional air. Each exhalation removes approximately the same volume as was inhaled.

Importance: Normal breathing involves rhythmic inhaling and exhaling. Therefore, we can quantify breathing using trigonometric equations. Changes in breathing, such as those due to increased activity, can then be measured.

Questions: How does lung capacity change over time? How does activity affect lung capacity?

Variables:

y

property exhibiting a rhythm (lung capacity)

t

time

A

amplitude

p

period

s y

phase shift in height of y

Methods: An apparatus called a spirometer is used to measure the volume of air inhaled and exhaled while breathing. A spirogram is the record of the changing volumes over time. The output of a spirogram can be quantified using trigonometric equations.

In general, we can describe any repeating rhythm by a sine wave:

where y is some property that exhibits a rhythm, t is time, A is the amplitude, p is the period, and s y is a phase shift in the height of y compared to the simplest sine wave (y = sin t where s y=0). s y can be thought of as the value of y half way between the peak and crest of the wave. The amplitude of a wave is the height from peak to crest. For the simplest sine wave, y = sin t, the amplitude is 2. The period of a wave is the time for completion of one cycle. For y = sin t, the period is 2p.

In humans, when each breath is completed, the lung still contains a volume of air, called the functional residual capacity (approximately 2200 mL in humans). Each inhalation adds from 500 mL of additional air for normal (resting) breathing. Each exhalation removes approximately the same volume as was inhaled. The volume of air inhaled and exhaled normally is called the tidal volume. It takes approximately 5 seconds to complete a single cycle. With this information, we can develop an equation for lung capacity as a function of time for normal (resting) breathing. Since s y describes the midpoint of the sine wave, we take the functional residual capacity plus 1/2 the amplitude to give us s y=2200+250=2450 mL. Our period is the the time for a single cycle, or p=5 seconds. The volume of air inhaled gives the amplitude, or A=1000 mL. We therefore have the following equation for normal breathing:

which we can graph.

As the degree of activity increases, the volume inhaled and exhaled increases from 500 up to 3000 and the functional residual volume decreases as greater amounts of air are exhaled. In our equation for lung capacity, this is reflected as an increase in amplitude (A) and a decrease in functional residual capacity.

Interpretation: As activity increases, a greater amount of air is forced in and out of the lungs. As a result, the reserve volume in the lungs decreases. However, the lungs have a limited total capacity of 5700-6200 mL. When inhalation and exhalation are at their maximum, the vital lung capacity is reached (approximately 4500-5000 mL). There is a residual volume remaining in the lungs (approximately 1200 mL), even with maximum exhalation, that cannot be forcibly expired.

Conclusions: We can use trigonometric equations to quantify changes in lung capacity over time. Quantifying lung capacity in the form of spirograms is useful for diagnosing illnesses or respiratory problems. Also, patterns in breathing are likely to differ among different groups of people and under differing circumstances.

Additional Questions:

1. Determine an equation for lung capacity as a function of time when breathing is forced at maximum and minimum exhalation. Assume the vital capacity is 4000 in order to determine amplitude. Assume the residual lung capacity is 1200 mL to determine s y. Assume a single breathing cycle is completed every 5 seconds.

2. During hyperventilation, breathing is very rapid with deep inhalations and exhalations. How would you modify the sine equation for lung capacity over time to reflect hyperventilation? How would a graph of hyperventilation compare to normal breathing?

3. Emphysema reduces the flexibility of alveolar walls in the lungs so that they are unable to recoil normally from expiration. This results in an increased residual volume. How would a spirogram of a person with emphysema compare to that of a healthy person, assuming the same total lung capacity? How would emphysema affect the vital capacity?

Sources: Berne, R. M. and M. N. Levy (editors). 1998. Physiology, 4 th edition. Mosby, Inc. St. Louis, MO.

Thibodeau, G. A. and K. T. Patton. 1996. Anatomy and Physiology, 3 rd edition. Mosby, Inc. St. Louis, MO.

Copyright 2000 M. Beals, L. Gross, S. Harrell

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RHYTHMS IN BREATHING AND LUNG CAPACITY Introduction: The respiratory system functions to filter, warm, and humidify the air we breathe, and to supply cells with oxygen while removing carbon dioxide. In animals that breathe with lungs, air moves into the lungs through the trachea and then back out again.

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When breathing, which we do everyday, we have a constant cycle of inhaling and exhaling. Because of this periodic functions can be implemented. You can use these functions to see the difference in breathing when doing different activities. For example, resting your period of breathing will be slower than respirating while doing a sport.

https://physoc.onlinelibrary.wiley.com/doi/10.1113/expphysiol.2005.032268

I wanted to see this application because we do it everyday and I wanted understand how breathing can be analyzed mathematically and see what uses it may have on my daily thoughts and actions.

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The seasons are brought around by a change in the length of days. This change happens due to the Earth's orbit around the Sun and the tilt of the Earth's axis. Because of the tilt, the northern hemisphere receives more light on one half of the orbit around the sun than the southern hemisphere. The sunlight in the north gradually decreases until it completes a full cycle around the sun and the daylight begins to increase once more. Because our orbit around the sun is a circle comparable to the London Eye; if we were to graph the length of days in one location, it would appear as a trigonometric function, the period of time for one cycle to be completed being one year.

https://www.hitchcockcenter.org/earth-matters/calculus-of-the-seasons/

I found this interesting because it is fascinating that something on such a large scale both in terms of time and space might fit such a simple graph, and something that you never though would be related to waves (the length of the day in a certain place) is graphed by them and repeating in exactly the same way year after year.

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(1) One of nature's most well-known periodic functions. Light flows through or reflects from many surfaces in different ways, but they are always periodic. The wavelength and frequency of color might vary, but they always travel in a wave-like pattern, exactly like a periodic function. They're also related to what we observed with the Ferris wheel function, because the wave's amplitude or width may fluctuate. This information may be useful in the future since (if I continue to study architecture), understanding how to alter light and colors is essential.
https://www.attune.com.au/2020/02/12/lets-talk-sound-frequencies/#:~:text=The%20total%20number%20of%20waves,or%205%20cycles%20per%20second.

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Kaori Garcia
1. Blood pressure can be represented in a periodic function with a minimum of 80 and maximum of 120 is the person is healthy. measures the pressure in the arteries when the heart contracts. The number 120 represents the pressure in the arteries when the heart contracts while 80 represents the pressure in the arteries as the heart relaxes between beats. Blood pressure problems arise when the minimum is less than 80 or when the maximum is more than 120. Overall, the application of periodic functions in blood pressure can usefully to detect any health and hearth issues.
2. https://courses.lumenlearning.com/precalctwo/chapter/modeling-with-trigonometric-equations/
3. I decided to learn about this application because I find the human body to be facinting. To me it is very interesting how just one number could determine is an individuals heart functions correctly.

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Andrés Martínez
1. Periodic functions can be used to observe the phases of the moon. This can be done because the phases repeat over the same period of time. We can use trigonometric functions to predict how the moon will look up in the sky. This is also useful to calculate the position of the earth in relation to the sun. The phases depend on the seasons, so we would see a pattern there. This is also useful because it gives us a perception of time. It can serve as some kind of natural clock.
2. http://www.mrbertman.com/projects/moon.pdf
3. I found this interesting because it is something fascinating that we don't realize it happens all the time. It is quite mind-boggling to stop and think for a moment that we are living in a floating rock traveling through the vastness of space.

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Sound is a periodic wave that travels by vibrating. Creating the thing that listens to each such as eardrums to vibrate at the same frequency and therefore hearing the sound that we hear. Because they are periodic waves the way they are represented is my using a periodic function, this way it shows how it works.

https://www.thermaxxjackets.com/sound-waves-and-characteristics/

I found that periodic functions can be used for more than a Ferris wheel specially in something I love such a listening, thanks to this I listen to my music which I am very grateful for and the graphs help me understand a little bit of how they work

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Sound Waves & Their Characteristics Explained

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What is a sound wave? Not being able to see a sound wave makes this a difficult question. Read our simple guide to learn all about sound waves.

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1Tides are very long-period waves that move through the oceans in response to the forces exerted by the moon and sun. Tides originate in the oceans and progress toward the coastlines where they appear as the regular rise and fall of the sea surface.
When the highest part, or crest, of the wave, reaches a particular location, high tide occurs; low tide corresponds to the lowest part of the wave, or its trough. The difference in height between the high tide and the low tide is called the tidal range.

2https://oceanservice.noaa.gov/facts/tides.html#:~:text=Tides%20are%20caused%20by%20gravitational%20pull%20of%20the%20moon%20and%20the%20sun.&text=Tides%20are%20very%20long%2Dperiod,fall%20of%20the%20sea%20surface.
3 I researched about tides because this application to trigonometry helped me notice how using calculations you can be able to determine or predict things that are going to happen

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Trigonometry is used in oceanography in calculating the height of tides in oceans. You can use trigonometry to graph the changes in high and low tides for a particular location.   For example, high tides and low tides can be modeled and predicted using periodic functions because scientists can determine the height of the water at different times of the day (when the water level is low, the tide is low).

https://www.mathnasium.com/real-life-applications-of-trigonometry

I found tidal gave s interesting because I think the ocean is very complex and I did not know you could use trigonometry to understand it better.

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1.- The motion of Earth can be described as rotation about an axis while undergoing circular motion about the Sun. Earth’s motion is repetitive. It takes approximately 24 hours for Earth to rotate through 360 ̊about its axis. One complete revolution about the sun takes approximately 365 and one third days, i.e., one solar year. Because the motion repeats itself, we refer to it as periodic motion. The requirement that the motion of an object must satisfy to be periodic is that it possess a definitive period. The period is the time that it takes for the object to travel through one complete cycle. In our example of Earth’s motion about the Sun, the period, T,of Earth’s circular motion is one solar year.
2.- http://engineering.nyu.edu/mechatronics/smart/pdf/experiments/Project4.pdf
3.- When I was brainstorming of thing in nature that could be periodic function, I began thinking of thing that made the same motion as a Ferris Wheel. That’s when Earth’s rotation around the sun came up as an example. I first googled if this would be correct and I was right. I feel like it’s really interesting that you can ring the specific periodic function of something that is so big and so important.

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The phases of the moon is an interesting way to look at periodic functions. The moon rotates around earth and depending on where the sun hits it it reflects more or less light however it always comes full circle starting new and going all the way around. Since the moon does not have its own light it relies on the sun to be seen. This to me is super interesting since, while it is true that I knew the moon changed the amount of light I did not really think about how it does go full circle with a followable pattern, the fact that our moon relies on the sun to shine on earth is also pretty cool. Periodic functions are everywhere however it might be hard to spot if you are not looking for them. That is why I believe we learn them, so that we can start understanding the world around us, to really feel what the things on earth are doing, and maybe after that we can start understanding the things going on in the vastness of space.
https://spaceplace.nasa.gov/moon-phases/en/
https://www.timeanddate.com/astronomy/moon/phases.html
sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwip5MODvOjzAhUhmWoFHToDDXoQwqsBegQIBxAB&url=https%3A%2F%2Fwww.khanacademy.org%2Fscience%2Fcosmology-and-astronomy%2Fearth-history-topic%2Fmoon-phases-and-eclipses%2Fv%2Fintro-to-moon-phases&usg=AOvVaw3OElVWNwt57JWPI-jKULoX

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Natalia Demesa
High tides and low tides can be modeled and predicted using periodic functions because scientists can determine the height of the water at different times of the day. For example, they measure when the level of the water is low or not, which will determine if the tide is low or not.
https://flexbooks.ck12.org/cbook/ck-12-interactive-algebra-2-for-ccss/section/4.7/primary/lesson/modeling-periodic-behavior-alg-2-ccss/
I found this particular example to be really interesting because it really helped me to get a better understanding of how periodic functions can be applied in real life.

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SOFIA MACHADO
The two types of stocks (cyclical and non-cyclical), always depend on the market and success of companies that come from consumer demand. There are two types of purchases, needs and wants. These different types of purchases determine the difference between cyclical and non-cyclical stocks. Cyclical stocks consistently rise and fall, whenever the economy is doing well people buy, and usually try to time when the market will go low to sell.
https://prezi.com/btjzvyf89spd/real-world-examples-of-periodic-functions/
I found this interesting because my dad is always checking the stock market, so I know a small amount about it, but I have seen it since I was young. I had never thought that periodic functions could be in real life, not to mention, something I have known for most of my life.