6.5 Graphs of Trigonometric Functions

8/4/2019 6.5 Graphs of Trigonometric Functions

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Transcendental

Functions6.5 Graphs of Trigonometric

Functions

RA Idoy

MATH17



• Graphs of Sine and Cosine

• Graphs of other sine waves



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



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



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.



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



2

2 3

2

2

2

3

2

2 O

Also referred as

sine wave



cos f t t

cos(t) is symmetric with respect

to the y-axis cos cost t



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




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.




Example:

Graph 3sin f t t



sin f t t

3sin f t t

Since the amplitude is 3,then we multiply all the ordinates

by 3.




Example:

Graph 2cos f t t



cos f t t

2cos f t t

Since the amplitude is -2,

then we multiply all the ordinates

by -2.




Example:

Graph 1

cos2

f t t



cos f t t

1

cos2

f t t

Since the amplitude is 1/2,

then we multiply all the ordinates

by 1/2.




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



sin f t t

This implies that one

cycle of the sine wave is

2π

1b 2

21

P



cos f t t

This implies that one

cycle of the sine wave is

2π

1b 2

21

P




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



cos2 f t t One cycle of f(t)=cos2t




Example:

Graph

Find the period:


the sine wave of f(t) is 4π.

1

cos2

f t t

2 24

1 / 2

Pb



1

cos

2

f t t One cycle of f(t)=cos(1/2)t




Example:

Graph

Find the period:


the sine wave of f(t) is 2/3π.

sin3 f t t

2 2

3

Pb



sin3 f t t

One cycle of f(t)=sin3t




Example:

Draw one cycle of

Find the period:


the sine wave of f(t) is 3π.

2

sin3

f t t

2 23

2 / 3

Pb



2

sin

3

f t t




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 _______




RECALL Example:




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




Example:

Graph

1sin4

f t t



1

sin4

f t t

*Graph of f(t)=sin t phase

shifts to the right by 1/4π units




Example:

Graph

1sin2

f t t



1

sin2

f t t

*Graph of f(t)=sin t phase

shifts to the left by 1/2π units




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




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

6.5 Graphs of Trigonometric Functions

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