Architecture

Trigonometric identities are also used in the field of architecture. Back in the ancient times architects had to be mathematicians because building designs were based upon mathematical principles specifically geometry and trigonometry. However, due to modern technology architects don’t have to be mathematicians. The technology provides the computations that architects would have used in their designs. The ideas of trigonometry are still used in the designs and construction of buildings (Salingaros, 1999).

A more specific example of architects using trigonometry is when they are trying to surveying an existing buildings. They can find measurements of heights, depths and widths from far away by noting the length and angle thee measurement was taken from. Then they will use trigonometric identities and properties to find measurements of heights, depths, and widths. This is known as measurements of a facade(Calter,2000).

Music Theory

Trigonometric identities are also used in music specifically describing the tones musical instruments can make. Sound waves are described mathematically as sine functions. Adjusting the frequency, period, or the phase shift of the sine function transforms the loudness, pitch, and tone of the sound. These ideas are applied in the composition of music but is not described in mathematical terms. Musicians can create more complex tones, which are known as harmonies in music, by using the sum of different sine functions (Quinn, et al., 2017).

The article “Music as math waves: exploring trigonometry through sound” mentioned different examples of using the sum of sines formula. A musician can create a tone of 400 Hz and an overtone of 800 Hz with the equation y=sin⁡(800πt)+sin⁡(1600πt). Computers are programed with these relationships of sine functions and sound waves to generate harmonies and tones. This is how ringtones are generated for phones (Quinn, et al., 2017).

Mechanical Engineering

In mechanical engineering engineers design machines and mechanisms to improve technology. Many of their designs involved circular motions and lateral movements and engineers must use trigonometry to describe these movements. An example of this is mentioned in an article by Sean Rule. He described an issue where a mechanical engineer had to use trigonometric identities to solve and represent the motion of a link attached to a circular part. The motion was represented by a triangle that is connected to a circle. Using the law of cosine and the Pythagorean identities, he determined the length of the link and the angle that would describe the rotation of the circular part. Engineers need the mathematical knowledge to describe how their designs work (Rule, 2006).