Trigonometry |
Basic trigonometry was covered in lessons 5-7 of Unit 6. The three trigonometric functions we covered were sine, cosine, and tangent. Sine is the defined as opposite/hypotenuse. Cosine is defined as adjacent/hypotenuse. Finally, tangent is defined as opposite/adjacent. Below are some real-life examples of trigonometry.
|
Trigonometry in Music Theory
In music and acoustics, a common way to measure amplitude is through sound waves. When you observe sound waves, it may look something like this:
Graph of a Sound Wave
http://www.mediacollege.com/audio/01/sound-waves.html
All sound wave graphs can be created through a basic trigonometric function, sine. The formula for a sound wave is f(x)=A(sin(k(x))). In this formula, A represents the frequency, or volume of a note. The higher the A value, the louder the note will be, and vice versa. Higher A values also make the crest of the wave larger, and the trough of the wave smaller. The next variable, k, determines the pitch of the note. The lower the k value, the lower the pitch will be. On a graph, higher k values will result in a shorter wavelength, raising the pitch. Trigonometric functions are applied on a daily basis on everything from listening to a song to simply having a conversation.
Trigonometry in Architecture
One of the most important topics in architecture is trigonometry. Architects use sine, cosine, and tangent to make precise angle and side calculations on their buildings. Buildings like the Louvre and the Golden Gate Bridge both use extensive amounts of trigonometry to achieve a high degree of perfection.
The Louvre Museum is constructed in the shape of a pyramid. I.M. Pei, the architect behind the design most likely used trigonometric ratios to calculate the length of the sides. If Pei knew the height and side length of the pyramid, then he could use tangent to calculate the length of the diagonal on the pyramid. On the Golden Gate bridge, the architect also would have used tan to find where the suspension wire needed to be placed on the length of the bridge.
Trigonometry in Navigation
Trigonometry can also be applied to navigation. For example, if you are a ship and are looking up at a lighthouse at a 45 degree angle and know how tall the lighthouse is, you can calculate how far you need to sail to reach the shore. You can also calculate this with the angle between you and the sun. Another application is an airplane that is decending to the ground. If you know the distance to the airport and the height at which you are at, then you can calculate the exact angle at which you must land.