# Sine Functions

## Contents

## Basic Description/Definition

A sine function is an trigonometric function defined by the relationship between a given angle in a right triangle and the ratio of the length of the side opposite that angle to the length of the hypotenuse. The sine model is commonly used to illustrate periodic or regular occurrences such as sound/light waves, temperatures, tides, etc. The graph of a sine function appears wave-like, with one wave segment repeated continuously over the x-axis. This is modeled by the image above, in which the blue arc that surrounds the green circle on the left is equal in value to the blue line on the graph to the right (from http://en.wikipedia.org/wiki/File:Sine_curve_drawing_animation.gif). The table above on the far right (from http://en.wikipedia.org/wiki/Sine) shows the values of the graph in 90 degree increments; these values will repeat over and over, so that when x = 450 degrees, sin (x) = 0, when x = 540 degrees, sin x = 1, etc. There are no specific horizontal or vertical asymptotes in the basic equation, and the end behavior depends on the values of A, B, C, and D in the general equation (see "General Formula and Vocabulary"), though the end behavior will never be constant in one direction because of the wave patterns that occur.

## General Formula and Vocabulary

The basic formula for a sine function is f(x) = A sin (B (x-C)) + D, where:

A = the amplitude, or ½ the distance between the maximum and minimum values of the function. Because it is a measure of distance, A will always be positive and can never equal zero. The higher the amplitude, the steeper and skinnier each wave will appear.

B = the frequency of the sine function, or the number of repeated segments over one completed cycle of the graph (ex. area of a beach covered by water over the course of three days--how many times did the tide go in and out?). Because it is a measure of number of repetitions, it can never be negative or equal to zero. The frequency of the function is closely related to its period, or the length of each repeated segment from beginning to end (i.e. the horizontal distance of each wave). The frequency times the period should equal the length of the completed cycle.

C = the horizontal shift of the function, or the shift of the wave left or right on the x-axis. Due to its repetitive nature, the horizontal shift in a sine graph is often inconsequential, and will not figure prominently in this article.

D = the vertical shift/midline of the function, or the upward/downward shift of the wave on the y-axis. This is calculated by adding the maximum and minimum values of the function and dividing by two, so that the distance between the minimum and the midline and the maximum and the midline is equal. A positive D value means that the graph is shifted upwards, and a negative D value means that it is shifted downwards, while a D value of zero means that it is centered on the x-axis itself.

Let’s look an example of a periodic function to get a better understanding of these terms and their relationships to each other...

Here’s what we can deduce about this function:

- This graph has a maximum value of 1 and a minimum value of -1. Therefore, the distance between the maximum and the minimum is 2.
- The amplitude is calculated by dividing the distance between the maximum and the minimum by two, so the amplitude of this function is 1.
- Assuming that the cycle started a 0 and ended at 360, there is 1 wave in the cycle. Therefore, the frequency of this function is 1.
- Because we know that the frequency times the period equals the duration of the cycle, and we’re assuming that the cycle is 360, the period must be 360 divided by 1, or 360.
- We know that the distance between the maximum and the minimum is 2, so in order to calculate the midline, we must divide 2 by 2 to get 1, then add 1 to the minimum (-1+1) or subtract 1 from the maximum (1 - 1) to get a midline at x = 0.

## Inverse Sine

The graph of inverse sine (a.k.a. arcsin) or f(x) = A sin^-1(B (x-C)) + D looks like the graph of sine rotated 90 degrees, (see below, left). However, when the graph is rotated 90 degrees, each x value has more than one y value, so it is no longer a function, but relation.

## Real Life Applications

Sine functions can be used to model a variety of real-life situations, such as:

- pulling a spring and releasing it
- electromagnetic, radio, or microwaves
- AC currents
- weather patterns/tides
- earthquakes, tsunamis
- astronomy and physics, especially in regards to location of objects in the universe as related to other objects
- heartbeats, brain waves
- business cycles/profitability
- virtually anything reoccurring and fairly constant