# Sine Functions

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 Revision as of 14:49, 18 November 2012 (edit)← Previous diff Revision as of 14:57, 18 November 2012 (edit) (undo)Next diff → Line 1: Line 1: {{Image Description {{Image Description - |ImageName=Where does a sine function come from? |Image=Sine curve drawing animation.gif |Image=Sine curve drawing animation.gif Line 13: Line 12: == Basic Description/Definition == == 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 (see image). 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 (see image). + + == 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. + [[Image:Sinexample.jpg]]

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# Teaching Materials

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## 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 (see image).

## 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.