Math 120: Elementary Functions Overview: Sine and Cosine Functions Review: 16 Point Circle The graphs of y=sin(x) and y=cos(x) Properties of y=sin(x) and y=cos(x) Boxing the “Wave” – the 4 quarters Sinusoid Functions amplitude frequency mid-line displacement phase shift Example: Fitting a sinusoid function to data

The 16-Point Unit Circle

graphs of sin(x) & cos(x) Be able to sketch the graphs of the sine and cosine functions and discuss domain & range periodicity (x-intercepts) symmetry relative maximums and minimums intervals of increase & decrease “Boxing the Wave”/the four quarters

Boxing the sine function

Boxing the cosine and sine functions

sinusoid functions A function is a sinusoid if it can be written in the form where a = amplitude = period c/b = phase shift (from starting point) d = (vertical) displacement from mid-line A function of the form is also a sinusoid

Specifics Given re-write as Example: Graph one period of Graph one period of

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)

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

Graph of plus data points