About trigonometric functions
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Introduction to sin and cos functions and graphing.

This topic is part of the TCS FREE high school mathematics 'How-to Library'. It introduces sine and cosine functions, and shows you how to draw their graphs with Maths Helper Plus.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

Sine and cosine functions have the form of a periodic wave:

The period, T, is the distance between any two repeating points on the function.

The amplitude, A, is the distance from the midpoint to the highest or lowest point of the function.

Phase shift is the amount of horizontal displacement of the function from its original position.

These periodic functions can be written in sine or cosine form.

The general form of the sine function is:

y = Asin(Bx + C) + D

where:

'A' is the amplitude of the function.

The period of the function is: T = 2p/B

The phase shift of the function is: C/B

A positive phase shift means the graph has moved to the left, while a negative phase shift means the graph has moved to the right.

Phase angle = (phase shift)*T/360. This is a common way to measure phase shift. A 360 phase shift is a shift through one complete period.

'D' is the amount of vertical displacement, or 'y' shift, of the mid point of the function above the 'x' axis.

The general form of the cosine function is:

y = Acos(Bx + C) + D

Cosine functions are identical to the sine functions, except that they are phase shifted to the left by 90. By accounting for this shift, any sine function can be written as a cosine function, and any cosine function can be written as a sine function.

Example: This example will work for both sine and cosine functions.

Use Maths Helper Plus to plot the sine function, given:

 amplitude 'A' = 1.5, period 'T' = p/2, phase shift = p/3 and 'y' shift = 0.5

Solution: Let the unknown sine function be: y = Asin(bx + c) + d

Then the period:

 

so:

Phase shift = c/b = p/3, so...

So the sine function to be graphed is: y = 1.5sin( 4x  + 4/3p ) + 0.5

Method:

Maths Helper Plus can graph sine or cosine functions easily. You just type the function then press the 'Enter' key to plot the graph.

For the 'sine' function, we have created a helpful document that calculates the equation when the amplitude, period, phase and 'y' shift are given. Also, if you know the equation:  y = asin(bx+c)+d, then it will calculate the phase shift, period, and other useful values. It shows all working steps and graphs the function as well.

 

Step 1 Download the free support file... We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Sine graph plotter.mhp'   File size: 7kb
Click here to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

Step 2  Display the parameters box

Press the F5 key to display the parameters box:

 

Click on the edit boxes and change the values to suit your problem type.

 

If 'X' = 1, then the function graphed is: y = Asin(Bx + C) + D, and the period and phase shift are calculated.

 

If 'X' = 2, then the edit box value are interpreted as follows:

 

'A' = amplitude,  'B' = period, 'T',  'C' = phase shift,  D = 'y' shift. The values for the equation of the sine function are calculated with working steps shown.

When you have entered your values, click the 'Update' button to refresh the calculations and graph display.

 

Step 3  Adjust the scale of the labelled diagram

You can change the scale horizontally by holding down Ctrl and pressing the left or right arrow key. Change the vertical scale by holding down Ctrl while pressing the up or down arrow key.

 

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