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when and engineer decides the heights, lengths of supports, the size, the weight, and the distance between each support trigonometry helps them calculate the angle each support should be at. Surveyors also use trigonometry to determine an unknown height or a distance without having to move. For example if they have to cut down a tree that is in the way they're not going to physically measure the whole tree they would use trigonometry to estimate the height without having to take an actually measurement.
Good, however this is an example of using triangles not one of periodic behaviour. - Ms. Payne

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When looking at sinusoidal functions you can see that it's a constant repition of the same wave lengths. An example is the hours in a day. In one day we always have 24 hours. It is a periodic relationship because the number of hours of daylight repeats each year, also the number of hours of daylight is based on the rotation of the earth which is also periodic. Therefore, the amount of hours in a day or the days in a year can be modeled by thus function.
Good example - Ms Payne

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Sound has many different factors that affect the height and width of a wave. Like any other waveform sound waves consist of wavelength, frequency, velocity, and amplitude. These numbers vary based on the air temperature; however, at sea level (68 degrees,) the speed of sound is 767 miles per hour. Sound waves are created through vibrations, music in general is a great example of a trigonometric function since sound waves can be used in sine waves. How loud and how high pitched the sounds are all factors of how wide and high the sine wave will be. Also sound waves are used in building soundproof rooms and houses in general.
Good example - Ms. Payne

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A sine or cosine can repeat itself over and over a period of time, but in music, the pitch can differ throughout a song. When your listening to music. In every note that determines the size of sine wave, in other words frequencies. If a note has a wide sine wave, then the volume has a low pitch. If a note has a narrow sine wave then has a high pitch. Volumes can determine the amplitude as well. If the amplitude is small the lower volume. If the amplitude is high the louder volume. Therefore soundwaves aren't repetitive, but have a same pattern. There are tons of related with music and other stuff including game development, building homes, engineering, etc... Trigonometry is one of important units to do in math, all of us use this almost everyday life, whether we know it or not.
Music is an excellen example. - Ms. Payne

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They complete the same action continuously that can be displayed as a sine function. You can model this by using one point on the wheel and follow the height it takes as the wheel turns.
Good, what else on the wheel relates to the graphical funciton? - Ms. Payne

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Someone skipping with a jump rope is a good example. Everytime the skipper bring the rope up and down it repeats the same cycle. The height of the rope from the ground can be measured with a sine function.
Neat example. - Ms. Payne

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Gps and cell phones rely on  triangulation and formulas involving sin and cos. Triangulation is a process that works by using the distance from two known points. A gps receiver measures the distance using the travel time of radio signals from satellites. For example, if somebody is between two cell towers on a highway, the position north and east of the first tower, and distance from the highway can be determined using sine and cosine. the cosine of the angle can be used to find the distance east of the tower. The sine of the angle can be used to find the distance north of the tower.
Good example, but this one is more related to triangles instead of a trig function. - Ms. Payne

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The rotation of the fan blades can be represented by a sine function. The length of the blade will represent the amplitude and the axle of the blade will represent the axis of the curve. The blade will spin showing the periodic nature of the sine function.
Great example. - Ms. Payne

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When you turn on a light switch or turn on the TV, you are using a trigonometric function. Modern day electrician companies send current down wires and use electric charge to reverse the direction to deliver power safely and reliably to homes and businesses. You can use trigonometry to model this flow and the change of direction, with the sine function used to model voltage.
As a physics teacher, I like this example! - Ms. Payne

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the rotation of one given point on a car tire can be modeled by a trigonometric function. as the car moves the tire will complete rotations causing the point to move upwards and downwards to the ground. The function can be changed if the size of the tire is increased or decreased
Good, but who wrote this?. - Ms. Payne

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A canoeist is circumnavigating a circular island on the St. Lawrence river. The distance between his two way radio and the radio held by his friend, who is sitting still at the campsite can be modeled by a sine function.  This model is important as the further he is the worse the reception is.  A period will be each time he passes the dock that his friend is next on.
Neat example. - Ms. Payne

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Like in music, sound waves travel in a repeating wave pattern, which can be represented graphically by sine and cosine functions. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. A graphical representation of music allows computers to create and understand sounds. Trigonometry plays an important role in speaker placement as well, since the angles of sound waves hitting the ears can influence the sound quality.
Excellent analysis - Sounds waves has times of silence which impact where noise needs to originate. - Ms. Payne

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A heart rate monitor, monitors your heart beat and shows the waves of every time it beats. A graphical representation of this helps your doctor to know one that your are alive and also it helps your doctor to tell if your heart rate is skipping a beat and also what is you maximum and minimum amount of beats your heart does. Periodic Behaviour is a great way of representing this because it is used to show behaviours that repeat.
Great example. - Ms. Payne

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Created by the gravitational pull between the moon and the sun, both tidal waves and tides are examples of periodic behaviour. Tidal waves happen because of daily tides; as the tides come in and out each day, they produce a rise and fall in sea level. High and low tides follow a periodic pattern that can be modeled with a sine function.
Great example. - Ms. Payne

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A rider enters the Ferris wheel and repeats the same motion for several cycles. The height of the rider, in reference to the ground or platform, can be modeled by a sine function.