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 difinitive. The effect of varying the coefficients on a sine or cosine graph will be investigated. The graph below shows

sin xover 720 degrees from -360 degrees to 360 degrees. This shows the familiar shape of the sine graph.The

x-axis will represent degrees andyvalue ofsin…orcos…. 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

sin and cosgraphs have a similar appearance. Thecosgraph is 90 degrees right along thex-axis. It has been moved +90 degrees along thex-axis. There maybe a rule to translate the one graph to equal the same values as the other.I used a spreadsheet to show the graphs. Initially I had problems as Excel was recognising

sin xandcos xusing radians and not degrees. 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 Pi which goes as follows:

=SIN(A2*PI()/180) …for Excel or…

Sin(x× Pi ÷ 80).

TransformationDescriptionTranslationThe graph/object has been movedfrom one set of co-ordinates to another. This movement is described using vectors [ ], withxat the top andyat the bottom.ReflectionThis is when there is a mirror image of the graph/object. The line of symmetry is given as y=, for exampley= 2x.RotationWhen a graph/object has been turned around it has beenrotated. The angle of rotation must be given as well as the centre of rotation, for example (2, 0) 90 degrees clockwise.EnlargementIf 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; ¼ or 1:4 .These diagrams show characteristics of coefficients

a,b,canddwherey=asin(bx+)c+All coefficients are positive and above 1.d.

KEY

- - - - -previous graph--------current graph-------- sine -------- cosine vertical spread of graph increases

oscillations increase graph transposed left graph transposed up Here are my final conclusions on

y=asin(bx+)c+, all points shown below are from my research and investigating.d

When the value of the coefficient is positive:

aincreases the vertical spread of the graph by its value

bincreases the oscillations per 360^{o}by its value

cmoves the graph left along the x-axis by its value

dmoves 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

bdecreases the oscillations per 360^{o}by its value

cmoves the graph left along the x-axis by < 1

dmoves the graph up the x-axis by < 1

When the value of the coefficient is negative:

ainverts the vertical spread of the graph by its value

bincreases the oscillations per 360^{o}and inverts the graph by its value

cmoves the graph right along the x-axis by its value

dmoves the graph down the x-axis by its value

Translatingsin xandcosx:

sin

=xcos(x-90)cos

=xsin(x+90)

To find:The peak of the graph

a+dThe trough of the graph

d-aOscillations per 360

^{o}b

Interesting facts associated sine:Normally cosine and sine are used to determine waves.

In music, if the frequency of two notes were

sin xandsin2xthey would be octaves.In music, if the frequency of two notes were

sin xandsin1.5xthey would be fifths.The formula for frequency is: speed/wavelength and measured in hertz (Hz).