On the axis below graph the Sine and Cosine Functions
FFuncFunctGraphs
y=sin x and y = cos x
x
y=sin x
y=cos x
0
π
2
π
3π
2
Domain:
Range:
That the shape of the sine and cosine curves forms a regular
pattern (the curve repeats after the wheel has gone around
once)? We say such curves are periodic. The period is the time
it takes to go through one cycle and then start over again.
2π
Mathematics REVIEW: y = Asin(B(x-C))
Period of a Sin/Cos Function:
A=
How does it affect the graph?
B=
How does it affect the graph?
C=
How does it affect the graph?
Amplitude
Phase
Physics Connections:
Frequency = f 
Angular
Frequency
Period
1 2
T 
f


2
State the amplitude, frequency and period of each of the functions graphed below.
Practice:
1. State the amplitude, frequency, & period of each:
x
a. y = sin(2x)
b. y = 2cos( )
2
2. On the axes from 0 to 2π, graph y=3cos(x).
3. On the axes from 0 to 2π, graph y=-2cos(x).
π
4. Graph y=sin(2(x+ ) in the interval 0 < x < 2π.
2
c. y = -3sin(3x)
5. Given the following equations, determine the amplitude, period, phase shift, and vertical shift
of each equation. Explain in sentences how you find each of them.
1 
6. a) On the same set of axes from 0 to 2π, graph y=3cos(2x) and y=sin  x  .
2 
1 
b) How many points satisfy the equation 3cos(2x)= sin  x  ? In words explain what those
2 
points represent?
7. a) On the same set of axes,
b) Find all values for which
3cos(x) - sin(2x) = 0
, graph y=3cos(x) and y=sin(2x) .
Practice
1. What is the period of the function y = 5 sin 3x?
(1) 5
(3) 3
2
2
(2)
(4)
5
3
2. A sound wave is modeled by the curve y  3 sin 4 x. What is the period of this curve?
(1) 
(3) 3

(2)
(4) 4
2
3. Given f(x) = 2sin5x and g(x) =2sin2x,
a. Which function has a higher frequency?
b. Which function has a greater period?
4. Given f(x) = 4cos3x and g(x) =4cos6x,
a. Which function has a higher frequency?
b. Which function has a greater period?
5. What happens to the period of the function if the frequency doubles?
6. What happens to the frequency if the period doubles?
6. Using the answers to questions 3 - 6, analyze the relationship between the period and
frequencies of the sine and cosine functions and support your concluding statement.
7. What is the amplitude of the function y 

2
2
(2)
3
(1)
2
sin 4 x ?
3
(3) 3
(4) 4
8. Which graph represents a sound wave that follows a curve whose period is π and that is in
the form y  a sin bx ?