Trigonometric Functions in Construction

Some of the many applications of trigonometry in construction include find the height of existing buildings with triangles, building trusses for roof support, and finding the desired roof pitch for a house.

GPS and cellphones rely on triangulation and formulas involving sin/cos.

Topic 3

Music is composed of waves of different frequencies and amplitudes and these can be described using sin/cos.

Topic 2

5 uses of trigonometric functions in real life

1) Electrical currents

2) Radio Broadcasting

3) Low and High Tides of the Ocean

4) Highways

5) Buildings

**Real Life Applications for Sine and Cosine Trigonometric Functions**

Topic 1

Trigonometry is especially important in architecture because it allows the architect to calculate distances and forces related to diagonal elements., for example on bridges and tall structures, the diagonal has to be strong and accurate to keep the structure standing.

Architecture

Space flight relies on calvulations and conversions to polat coordiates....

because they help model orbital motions. Polar coordinates express a position on a two-dimensional plane using an angle from a fixed direction and a distance from a fixed point. Polar coordinates can be converted to Cartesian coordinates- the coordinate plane that we are used to seeing, and have been seeing since elementary. Polar coordinates can be converted to the Cartesian coordinates (x,y) by using sine and cosine functions. By multiplying the polar coordinates by cosine, the x coordinate can be obtained. By multiplying the polar coordinates by sine the y coordinate can be found.

If a trumpet sounds at 440 Hz, at various amplitudes, the summation of sine waves or in other words Fourier series will be 440 Hz, 880 Hz, 1, 320 Hz, 1, 760 Hz.

As we know sound travels in waves and frequencies.

A French scientist and mathematician by the name of Jean Baptiste Fourier proved that any waveform that repeats itself after a period of time (such as a musical sound) can be expressed as the sum of an infinite set of sine curves. As we know sound travels in waves and frequencies.

Ballisic trajectories rely on sin/cos

Triangulation a process that works by using the distance from two known points, is used in cell phones equipped with GPS. A GPS receiver measures distance using the travel time of radio signals from satellites.

For example,

If someone is between two cell phone 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

In other words, the path of motion of an object that is shot, thrown, flung, kicked, etc. can be broken up into its x and y components of the object's starting speed.

v0x=v0cos(θ)

v0y=v0sin(θ)

With these components we can find he displacement or the distance that the object traveled , we can also find the time of the motion, and the object's maximum height

Jessica Lovius

7th Period

Pre- Cal

Essay

Trigonometric functions- believe it or not can be found almost everywhere around us. Whether you are listening to music, or looking at a skyscraper, sine and cosine can be found in all walks of life. The sine and cosine functions can also be represented in 90 degree triangles all around us. For example, the distance of shadows of a tree and a person’s height can be associated with a right triangle. I researched the topics of space flight and polar coordinates, sound waves, ballistic trajectory, and GPS in cell phones to examine real- world applications of sine and cosine.

Space flight relies on calculations and conversions to polar coordinates because they help model orbital motions. Polar coordinates express a position on a two-dimensional plane using an angle from a fixed direction and a distance from a fixed point. Polar coordinates can be converted to Cartesian coordinates- the coordinate plane that we are used to seeing, and have been seeing since elementary. Polar coordinates can be converted to the Cartesian coordinates (x,y) by using sine and cosine functions. By multiplying the polar coordinates by cosine, the x coordinate can be obtained. By multiplying the polar coordinates by sine the y coordinate can be found.

As mentioned before, sounds and music travels in waves. Depending on the note that is played, the amplification of the wave, and thus the sine curve changes as the musical note changes. I also learned that a French scientist and mathematician by the name of Jean Baptiste Fourier proved that any waveform that repeats itself after a period of time can be expressed as the sum of an infinite set of sine curves. A ballistic trajectory or in other words the path of an object in projectile motion describes an object that has been shot, thrown, flung, kicked, etc. can be broken up into its x and y components of the object's starting speed. When breaking up the initial/starting velocity, we use sine and cosine functions. With these components we can find the displacement or the distance that the object traveled , we can also find the time of the motion, and the object's maximum height with the sine and cosine functions.

Triangulation a process that works by using the distance from two known points, is used in cell phones equipped with GPS. A GPS receiver measures distance using the travel time of radio signals from satellites. Sine and cosine functions are used to find the location and distances in the GPS system of the cell phone. I have used sine and cosine functions in my physics class, but with this project I have also observed additional applications of sine and cosine functions. In real life, sine and cosine functions can be used in space flight and polar coordinates, music, ballistic trajectories, and GPS and cell phones.

I end off with a quote by Jean Baptist- Joseph Fourier which says, " The deep study of nature is the most fruitful source of mathematical discoveries."