Mathematics: Sine and cosine coefficients
Here is a sample of a past piece of coursework, studying the effects on sine and cosine graphs. This GCSE coursework is quite old and lacks mathematical reasoning. Please do not take the information you read here as definitive.
The effect of varying the coefficients on a sine or cosine graph will be investigated. The graph below shows sine x over 720° from -360° to 360°. This shows the familiar shape of the sine graph.
The x-axis will represent degrees and y value of sine or cosine. This is the standard manner of setting out graphs and will save confusion. Red will indicate sine and green will represent cosine.
I have noticed that both sine and cosine graphs have a similar appearance. The cosine graph is 90° right along the x-axis. It has been moved +90° along the x-axis. There may be a rule to translate the one graph to equal the same values as the other.
I used a spread sheet to show the graphs. Initially I had problems as Excel was recognising sine x and cosine x using radians and not degrees. In passing, it is worth mentioning this text has been copied from Nahoo at nahoo.net. I will be using degrees so a formula was needed to convert the radians.
I found a formula for translating radians into degrees using the help facility on Excel. There was a formula using π which goes as follows:
=SIN(A2*PI()/180) …for Excel or… sin(x × π ÷ 80).
|Translation||The graph/object has been moved from one set of co-ordinates to another. This movement is described using vectors [ ], with x at the top and y at the bottom.|
|Reflection||This is when there is a mirror image of the graph/object. The line of symmetry is given as y =, for example y = 2x.|
|Rotation||When a graph/object has been turned around it has been rotated. The angle of rotation must be given as well as the centre of rotation, for example (2, 0) 90° clockwise.|
|Enlargement||If a graph/object is made larger, smaller, stretched or squeezed it is enlargement. Enlargement can be written as fractions or as a ratio, for example; Ľ or 1:4.|
These diagrams show characteristics of coefficients a, b, c and d where y = a sin (b x + c) + d. All coefficients are positive and greater than or equal to 1.
Here are my final conclusions on y = a sin ( b x + c) + d, all points shown below are from my research and investigating.
When the value of the coefficient is positive:
- a increases the vertical spread of the graph by its value
- b increases the oscillations per 360° by its value
- c moves the graph left along the x-axis by its value
- d moves the graph up the x-axis by its value
When the value of the coefficient is less than 1:
- a decreases the vertical spread of the graph by its value
- b decreases the oscillations per 360° by its value
- c moves the graph left along the x-axis by < 1
- d moves the graph up the x-axis by < 1
When the value of the coefficient is negative:
- a inverts the vertical spread of the graph by its value
- b increases the oscillations per 360° and inverts the graph by its value
- c moves the graph right along the x-axis by its value
- d moves the graph down the x-axis by its value
Translating sine x and cosine x:
- sin x = cos ( x-90)
- cos x = sin ( x+90)
- The peak of the graph a + d
- The trough of the graph d – a
- Oscillations per 360°b
Interesting facts associated sine:
- Normally cosine and sine are used to determine waves.
- In music, if the frequency of two notes were sin x and sin 2 x they would be octaves.
- In music, if the frequency of two notes were sin x and sin 1.5 x they would be fifths.
- The formula for frequency is: speed/wavelength and measured in hertz (Hz).
I have updated the charts to use vector graphics and corrected some of the obvious errors. However, it has been many years since I wrote this small project, but comments are still welcome.