Math 120: Elementary FunctionsOverview: Sine and Cosine Functions

Review: 16 Point CircleThe graphs of y=sin(x) and y=cos(x)Properties of y=sin(x) and y=cos(x)Boxing the “Wave” – the 4 quartersSinusoid Functions

amplitudefrequencymid-line displacementphase shift

Example: Fitting a sinusoid function to data

The 16-Point Unit Circle

1,0

3 1,2 2

31 ,2 2 1

2, 1

2

0,1

1,0

0, 1

31 ,2 2

31 ,2 2

31 ,2 2

12 2, 1

2 2,1 1

1 12 2,

3 1,2 2

3 1,2 2

3 1,2 2

graphs of sin(x) & cos(x)

Be able to sketch the graphs of the sine and cosine functions and discuss

domain & rangeperiodicity (x-intercepts)symmetry relative maximums and minimumsintervals of increase & decrease“Boxing the Wave”/the four quarters

Boxing the sine function

2 3

2 2

1

1

mid line

amplitude

starting point of the "wave"

4 quarters

one period

Boxing the cosine and sine functions

2 3

2 2

1

1

sinusoid functions

A function is a sinusoid if it can be written in the form

where a = amplitude = periodc/b = phase shift (from starting point)d = (vertical) displacement from mid-line

A function of the form is also a sinusoid

sin( )y a b x c d

2 b

cosy a b x c d

sin cy a b x db

Specifics

Given re-write as

Example: Graph one period of

Graph one period of

siny a bx c d

sin cy a b x db

:a amplitide2 : periodb :

: ; :

c phaseshiftbleft right

:d midlinedisplacement

2sin 3 12

y x

3cos 2 14

y x

modeling periodic behavior with sinusoids

Given max/min values M and m, find

the mid-line amplitude

period

starting point (a.k.a. phase shift)

2M md

2M ma

2" "

bone wave

cb

Examples

Given the normal monthly temperature in Helena MT, model the temperature as a sinusoid function

Max M = 68, min m = 20, period = 12, mid-line displacement = 44, amplitude = 24, phase shift = 4

Ans:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Temp 20 26 35 44 53 61 68 67 56 45 31 21

224sin 4 44 24sin 446 6 3

y x x

Graph of plus data points

2 4 6 8 1 0 1 2

1 0

2 0

3 0

4 0

5 0

6 0

7 0

24sin 4 446

y x