More Real Life Relations of Sine, Cosine, and other Circular Functions

The Fundamental Circular Functions can be also used in:



Surveying

This graph also shows that while the satisfaction for "providing educational opportunities is low", the importance in the customers' eyes is very low as well. Therefore, ABC Company now has the knowledge needed to allocate their dollars to the most important attributes.

Graphing, can be used in illustrating bacterial growth (as I remember, we studied this)

Calculation of generation time

he exactly doubled points from the absorbance readings were taken and, the points were extrapolated to meet the respective time axis.

Generation Time =   (Time in minutes to obtain the absorbance 0.4) – (Time in minutes to obtain the absorbance 0.2)

                           = 90-60

                           = 30 minutes

Let No = the initial population number

Nt     =   population at time t

N      =     the number of generations in time t

Sine is also used in Pendulum Motion. 

The position of the pendulum bob (measured along the arc relative to its rest position) is a function of the sine of the time.

It is used in signal transmission, television, radio broadcasting involves waves described in sine/cosine.

Diagram of the electric fields (E) and magnetic fields (H) of radio waves emitted by a monopole radio transmitting antenna (small dark vertical line in the center). The E and H fields are perpendicular as implied by the phase diagram in the lower right.

Lastly, it is also used in graphing stock exchanges.

The Philippines Stock Market (PSEi) decreased to 7230.55 Index points in December from 7294.37 Index points in November of 2014. Stock Market in Philippines averaged 2363.05 Index points from 1986 until 2014, reaching an all time high of 7392.20 Index points in May of 2013 and a record low of 129.52 Index points in February of 1986.

Many compression algorithms, like JPEG, GIF, JPG (and other picture formats) use fourier transforms that rely on sine and cosine. In fact most anything involving sound waves will rely on sin/cos. Space flight relies on calculations and conversions to polar coordinates. So do satellites. Ballistic trajectories rely on sin/cos, and there are numerous other uses of them in physics. 

Sine and Cosine in Real Life

The earliest studies of triangles can be traced to the 2nd millennium BC, first as Egyptian Mathematics and Babylonian Mathematics. The first systematic study of trigonometric functions began in Hellenistic Mathematics.

Sine is one of the six fundamental circular functions of an angle. Etymologically, the word sine derives from a Sanskrit word jiva, in Arabic, jaib, meaning "bosom", "bay" or "chord". The word sine was first introduced in 1950's.

Sin(x) as graphed

Animation showing how the sine function (in red)  is graphed from the y-coordinate (red dot) of a point on the unit circle (in green) at an angle of θ in radians.

In Triangle, with angle θ, sine is calculated as:

sin(θ) = Opposite / Hypotenuse

In picture form: 

For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.

Cosine, is also one of the six fundamental circular functions. In fact, the sine and cosine functions are closely related and can be expressed in terms of each other.

The cosine definition basically says that, on a right triangle, the following measurements are related:

• the measurement of one of the non-right angles (q)
• the length of the side adjacent to that angle
• the length of the triangle's hypotenuse

Cos (x), as graphed

In triangle:

Relating in real life:

You might ask, "Why do we study this?" "Is this essential in real life?" Of course, yes. Sinusoidal waves (or sine waves for short) have turned out to be essential to understanding how our world works. 

One example is sound. You might not know that we use the sine and cosine functions while you play instruments, or listen to stereo. You listen to sound waves. We can think of these as having the shape of sine waves. For example, if you know anything about playing a piano, the note A above middle C produces a wave shaped like . If you figure out the period of this function (using the theorem from class) you'll see that this wave has 440 complete cycles every second.

The graph of Note A: 

The graph of Note C#:

The graph of Note E:

But when you listen to your stereo sound, you hear a lot more. You hear  more than one note at a time. It's simple, it is the combination of the sine functions of the Note A, C#, and E. 

You can see that the graph is no longer as sine curve, but there is a pattern to it. And the pattern only repeats, so it is still a periodic function.