6.5 graphs of trigonometric functions
TRANSCRIPT
![Page 1: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/1.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 1/37
Transcendental
Functions6.5 Graphs of Trigonometric
Functions
RA Idoy
MATH17
![Page 2: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/2.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 2/37
• Graphs of Sine and Cosine
• Graphs of other sine waves
![Page 3: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/3.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 3/37
The y-axis is now
the f(t) axis
The x-axis is now
the t-axis
The tick marks for the t-axisare rational multiple of π
2
2 3
2
2
3
2
2
![Page 4: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/4.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 4/37
2
2 3
2
2
3
2
2
2
3
2
2 O
As t goes from 0 to π/2
sin(t) goes from 0 to 1
As t goes from π/2 to π sin(t) goes from 1 to 0
As t goes from π to 3π/2
sin(t) goes from 0 to -1
As t goes from 3π
/2 to 2π
sin(t) goes from -1 to 0
![Page 5: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/5.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 5/37
2
2 3
2
2
2
3
2
2 O
sine curvesine wave
The portion of a graph
over one period is called a cycle.
![Page 6: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/6.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 6/37
2
2 3
2
2
3
2
2
2
3
2
2 O
As t goes from 0 to π/2
cos(t) goes from 1 to 0
As t goes from π/2 to π
cos(t) goes from 0 to -1
As t goes from π to 3π/2
cos(t) goes from -1 to 0
As t goes from 3π/2 to 2π
Cos(t) goes from 0 to 1
![Page 7: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/7.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 7/37
2
2 3
2
2
2
3
2
2 O
Also referred as
sine wave
![Page 8: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/8.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 8/37
cos f t t
cos(t) is symmetric with respect
to the y-axis cos cost t
![Page 9: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/9.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 9/37
Graphs of Other Sine Waves
Other sine waves are obtained
from functions defined by the
equations of the form
sin
cos
f t a b t c
f t a b t c
![Page 10: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/10.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 10/37
Graphs of Other Sine Waves
Let’s start with a for amplitude.
Definition:The maximum value of f(t) is
|a| and the minimum value is
|-a|. The number |a| is calledthe amplitude of the sine wave.
![Page 11: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/11.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 11/37
Graphs of Other Sine Waves
Example:
Graph 3sin f t t
![Page 12: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/12.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 12/37
sin f t t
3sin f t t
Since the amplitude is 3,then we multiply all the ordinates
by 3.
![Page 13: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/13.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 13/37
Graphs of Other Sine Waves
Example:
Graph 2cos f t t
![Page 14: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/14.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 14/37
cos f t t
2cos f t t
Since the amplitude is -2,
then we multiply all the ordinates
by -2.
![Page 15: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/15.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 15/37
Graphs of Other Sine Waves
Example:
Graph 1
cos2
f t t
![Page 16: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/16.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 16/37
cos f t t
1
cos2
f t t
Since the amplitude is 1/2,
then we multiply all the ordinates
by 1/2.
![Page 17: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/17.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 17/37
Graphs of Other Sine Waves
Theorem:
The period P of a periodic
function defined by either
where b≠0 is given by:
.
sin
cos
f t bt
f t bt
2P
b
![Page 18: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/18.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 18/37
sin f t t
This implies that one
cycle of the sine wave is
2π
1b 2
21
P
![Page 19: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/19.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 19/37
cos f t t
This implies that one
cycle of the sine wave is
2π
1b 2
21
P
![Page 20: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/20.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 20/37
Graphs of Other Sine Waves
Example:
Graph
Find the period:
This implies that one cycle of
the sine wave of f(t) is π.
cos2 f t t
2 2
2
Pb
![Page 21: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/21.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 21/37
cos2 f t t One cycle of f(t)=cos2t
![Page 22: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/22.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 22/37
Graphs of Other Sine Waves
Example:
Graph
Find the period:
This implies that one cycle of
the sine wave of f(t) is 4π.
1
cos2
f t t
2 24
1 / 2
Pb
![Page 23: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/23.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 23/37
1
cos
2
f t t One cycle of f(t)=cos(1/2)t
![Page 24: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/24.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 24/37
Graphs of Other Sine Waves
Example:
Graph
Find the period:
This implies that one cycle of
the sine wave of f(t) is 2/3π.
sin3 f t t
2 2
3
Pb
![Page 25: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/25.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 25/37
sin3 f t t
One cycle of f(t)=sin3t
![Page 26: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/26.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 26/37
Graphs of Other Sine Waves
Example:
Draw one cycle of
Find the period:
This implies that one cycle of
the sine wave of f(t) is 3π.
2
sin3
f t t
2 23
2 / 3
Pb
![Page 27: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/27.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 27/37
2
sin
3
f t t
![Page 28: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/28.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 28/37
Graphs of Other Sine Waves
RECALL:
Polynomial function defined by
f(x)=a(x+c)n
if c<0:
Shift f(x)=axn to the _______
if c>0:Shift f(x)=axn to the _______
![Page 29: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/29.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 29/37
Graphs of Other Sine Waves
RECALL Example:
![Page 30: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/30.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 30/37
Graphs of Other Sine Waves
The same shifting is observed in
sine and cosine functions.
|c| is called the phase shift of the
graph
sin
cos
f t t c
f t t c
sin
cos
f t t c
f t t c
*Graph of f(t)=sin t or
f(t)= cos t shifts
to the right by c units
*Graph of f(t)=sin t or
f(t)= cos t shifts
to the left by c units
![Page 31: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/31.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 31/37
Graphs of Other Sine Waves
Example:
Graph
1sin4
f t t
![Page 32: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/32.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 32/37
1
sin4
f t t
*Graph of f(t)=sin t phase
shifts to the right by 1/4π units
![Page 33: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/33.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 33/37
Graphs of Other Sine Waves
Example:
Graph
1sin2
f t t
![Page 34: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/34.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 34/37
1
sin2
f t t
*Graph of f(t)=sin t phase
shifts to the left by 1/2π units
![Page 35: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/35.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 35/37
Graphs of Other Sine Waves
Example:
Graph
Solution:
since |a| = 1/2, amplitude is1/2
since b = 1, period is 2π since |c| is π, f(t)=1/2sint isshifted π units to the left
1
sin2
f t t
![Page 36: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/36.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 36/37
![Page 37: 6.5 Graphs of Trigonometric Functions](https://reader034.vdocuments.us/reader034/viewer/2022051202/577d25aa1a28ab4e1e9f5fe4/html5/thumbnails/37.jpg)
8/4/2019 6.5 Graphs of Trigonometric Functions
http://slidepdf.com/reader/full/65-graphs-of-trigonometric-functions 37/37
Graphs of Other Sine Waves
Example:
Graph
Solution:
since |a| = 4, amplitude is 4
since b = 2, period is 2π/2 = π
since |c| is π/4, f(t)=4cos2tis shifted π/4 units to theright
1
4cos24
f t t