Answer to Question #103432 in Trigonometry for tamara

Question

What other phenomena can be modeled using sinusoidal functions. ?

Answer 1

Sinusoidal functions can be used to model periodic phenomena that do not involve angles as the independent variables.The amplitude ,phase shift ,period, and vertical shift of the basic sine or cosine function can be adjusted to fit the characteristics of the phenomena being modelled.

Now that we have completed our study of sinusoidal functions, we can use our knowledge to apply it to the real world around us. We should now understand that any variable that is cyclical, harmonic, oscillating, or periodic in nature can be modeled graphically by a sine or cosine wave. There are countless applications of sinusoidal modeling in real life. Some of these applications include:

-Changes in Temperature over time

-Hours of daylight over time

-Population growth/decay over time

-Ocean wave heights (high and low tides) over time

-Sound waves -Biorhythm waves

-Electrical currents

-Ferris wheels and roller coasters

-Tsunamis and tidal waves

-Earthquakes

-Wheels, Trampolines, Swings

Example

Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.

Determining the Amplitude and Period of a Sinusoidal Function

Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function. The amplitude of a sinusoidal function is the distance from the mid-line to the maximum value, or from the mid-line to the minimum value. The mid-line is the average value. Sinusoidal functions oscillate above and below the mid-line, are periodic, and repeat values in set cycles.that the period of the sine function and the cosine function is $2\Pi$ . In other words, for any value of $sin(x2\pi k)=sinx$ or $cos(x2\pi k)=cosx$ where k is an integer.