Transcript PPT #1
Graphs of the
Sine and Cosine
Functions
Graphing Trigonometric
Functions
Graph in xy-plane
Write functions as
y
y
y
y
y
y
=
=
=
=
=
=
f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
=
=
=
=
=
=
sin x
cos x
tan x
csc x
sec x
cot x
Variable x is an angle, measured in radians
Can be any real number
Graphing the Sine Function
Periodicity: Only need to graph on
interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Sine Function
Domain: All real numbers
Range: [{1, 1]
Odd function
Periodic, period 2¼
x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
y-intercept: 0
Maximum value: y = 1, occurring at
Minimum value: y = {1, occurring at
Transformations of the Graph
of the Sine Functions
Example.
Problem: Use the graph of y = sin x to
graph
Answer:
Graphing the Cosine Function
Periodicity: Again, only need to graph
on interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Cosine Function
Domain: All real numbers
Range: [{1, 1]
Even function
Periodic, period 2¼
x-intercepts:
y-intercept: 1
Maximum value: y = 1, occurring at
x = …, {2¼, 0, 2¼, 4¼, 6¼, …
Minimum value: y = {1, occurring at
x = …, {¼, ¼, 3¼, 5¼, …
Transformations of the Graph
of the Cosine Functions
Example.
Problem: Use the graph of y = cos x to
graph
Answer:
4
2
3
2
2
-2
-4
2
2
5
2
3
Sinusoidal Graphs
Graphs of sine and cosine functions
appear to be translations of each other
Graphs are called sinusoidal
Conjecture.
Amplitude and Period of
Sinusoidal Functions
Graphs of functions y = A sin x and y
= A cos x will always satisfy inequality
{jAj · y · jAj
Number jAj is the amplitude
Amplitude and Period of
Sinusoidal Functions
Graphs of functions y = A sin x and
y = A cos x will always satisfy
inequality {jAj · y · jAj
Number jAj is the amplitude
4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions
Period of y = sin(!x) and
y = cos(!x) is
4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions
Cycle: One period of y = sin(!x) or
y = cos(!x)
4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions
Cycle: One period of y = sin(!x) or
y = cos(!x)
Amplitude and Period of
Sinusoidal Functions
Theorem. If ! > 0, the amplitude and
period of y = Asin(!x) and
y = Acos(! x) are given by
Amplitude = j Aj
Period =
.
Amplitude and Period of
Sinusoidal Functions
Example.
Problem: Determine the amplitude and
period of y = {2cos(¼x)
Answer:
Graphing Sinusoidal Functions
One cycle contains four important
subintervals
For y = sin x and y = cos x these are
Gives five key points on graph
Graphing Sinusoidal Functions
Example.
Problem: Graph y = {3cos(2x)
Answer:
4
2
3
2
2
-2
-4
2
2
5
2
3
Finding Equations for
Sinusoidal Graphs
Example.
Problem: Find an equation for the graph.
6
Answer:
4
2
3
5
2
2
3
2
3
2
2
-2
-4
-6
2
2
5
2
3
Key Points
Graphing Trigonometric Functions
Graphing the Sine Function
Properties of the Sine Function
Transformations of the Graph of the
Sine Functions
Graphing the Cosine Function
Properties of the Cosine Function
Transformations of the Graph of the
Cosine Functions
Key Points (cont.)
Sinusoidal Graphs
Amplitude and Period of Sinusoidal
Functions
Graphing Sinusoidal Functions
Finding Equations for Sinusoidal
Graphs
Graphs of the
Tangent, Cotangent,
Cosecant and Secant
Functions
Section 5.5
Graphing the Tangent
Function
Periodicity: Only need to graph on
interval [0, ¼]
Plot points and graph
Properties of the Tangent
Function
Domain: All real numbers, except odd
multiples of
Range: All real numbers
Odd function
Periodic, period ¼
x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
y-intercept: 0
Asymptotes occur at
Transformations of the Graph
of the Tangent Functions
Example.
Problem: Use the graph of y = tan x to
8
graph
6
Answer:
4
2
3
2
2
-2
-4
-6
-8
2
2
5
2
3
Graphing the Cotangent Function
Periodicity: Only need to graph on
interval [0, ¼]
Graphing the Cosecant and
Secant Functions
Use reciprocal identities
Graph of y = csc x
Graphing the Cosecant and
Secant Functions
Use reciprocal identities
Graph of y = sec x
Key Points
Graphing the Tangent Function
Properties of the Tangent Function
Transformations of the Graph of the
Tangent Functions
Graphing the Cotangent Function
Graphing the Cosecant and Secant
Functions
Phase Shifts;
Sinusoidal Curve
Fitting
Graphing Sinusoidal Functions
y = A sin(!x), ! > 0
Amplitude jAj
Period
y = A sin(!x { Á)
Phase shift
Phase shift indicates amount of shift
To right if Á > 0
To left if Á < 0
Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
Determine amplitude jAj
Determine period
Determine starting point of one cycle:
Determine ending point of one cycle:
Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
Divide interval
into four
subintervals, each with length
Use endpoints of subintervals to find the
five key points
Fill in one cycle
Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
Extend the graph in each direction to
make it complete
Graphing Sinusoidal Functions
Example. For the equation
(a) Problem: Find the amplitude
Answer:
(b) Problem: Find the period
Answer:
(c) Problem: Find the phase shift
Answer:
Finding a Sinusoidal Function
from Data
Example. An experiment in a wind tunnel generates
cyclic waves. The following data is collected for 52
seconds.
Let v represent the wind speed in feet per second and
let x represent the time in seconds.
Time (in seconds), x
Wind speed (in feet per second), v
0
21
12
42
26
67
41
40
52
20
Finding a Sinusoidal Function
from Data
Example. (cont.)
Problem: Write a sine equation that
represents the data
Answer:
Key Points
Graphing Sinusoidal Functions
Finding a Sinusoidal Function from
Data