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Understanding Trigonometric Graphs: Sine, Cosine, Tangent, Secant, and Amplitude - Prof. C, Study notes of Trigonometry

Harford Community CollegeTrigonometry
Prof. Christopher W. Jones

An in-depth exploration of various trigonometric graphs, including sine, cosine, tangent, secant, and their respective amplitudes, periods, and vertical translations. Learn how to plot these functions using tables of values and understand their key features.

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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bg1
4.1 & 4.2 – Basic Graphs and Amplitude
Sine and Cosine Graphs
- Plot y = sin(x) and y = cos(x) on separate graphs using a table of values
Domain and Range
- The domain of a graph is the set of all x-values a graph may assume
- The range of a graph is the set of all y-values a graph may assume.
Amplitude of Sine/Cosine
- The amplitude is a measure of the “height” of the trigonometric graph.
Let
)cos(
)sin(
xAy
xAy
. Then A is the amplitude of the graph.
Examples: Plot y = 3sin(x) and y = 4cos(x) on separate graphs using a table of values.
Compare them to y = sin(x) and y = cos(x).
Examples: Plot y =
)sin(
2
1x
and y = 3cos(x)
20 x
on separate graphs using rules
amplitude. Compare them to y = sin(x) and y = cos(x).
Period
- The period of a trigonometric function is the distance a function takes to complete
one full cycle
Let
)cos(
sin
Bxy
Bxy
. Then
B
2
is the period of the graph.
Examples: Plot y = sin(4x) and y = cos
x
2
1
40 x
on separate graphs using a table
of values. Compare them to y = sin(x) and y = cos(x).
Examples: Plot y = 2sin
x
3
1
and y =
where
60 x
on separate graphs
using rules of amplitude and period. Compare them to y = sin(x) and y = cos(x).
Vertical Translations
- A vertical translation moves the shape of the graph up or down.
- Let
cxy
cxy
)cos(
)sin(
. Then move the graph up/down c units
Examples: Plot y = sin(x) + 2 and y = cos(x) - 3
20 x
on separate graphs using a
table of values. Compare them to y = sin(x) and y = cos(x).
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Partial preview of the text

Download Understanding Trigonometric Graphs: Sine, Cosine, Tangent, Secant, and Amplitude - Prof. C and more Study notes Trigonometry in PDF only on Docsity!

4.1 & 4.2 – Basic Graphs and Amplitude

Sine and Cosine Graphs

- Plot y = sin(x) and y = cos(x) on separate graphs using a table of values

Domain and Range

- The domain of a graph is the set of all x-values a graph may assume

- The range of a graph is the set of all y-values a graph may assume.

Amplitude of Sine/Cosine

- The amplitude is a measure of the “height” of the trigonometric graph.

Let

cos( )

sin( )

y A x

y A x

. Then A is the amplitude of the graph.

Examples: Plot y = 3sin(x) and y = 4cos(x) on separate graphs using a table of values.

Compare them to y = sin(x) and y = cos(x).

Examples: Plot y =

sin( )

x

and y = 3cos(x) 0  x  2 on separate graphs using rules

amplitude. Compare them to y = sin(x) and y = cos(x).

Period

- The period of a trigonometric function is the distance a function takes to complete

one full cycle

Let

cos( )

sin

y Bx

y Bx

. Then

B

is the period of the graph.

Examples: Plot y = sin(4x) and y = cos

x

0  x  4 

on separate graphs using a table

of values. Compare them to y = sin(x) and y = cos(x).

Examples: Plot y = 2sin

x

and y =

cos( 4 )

x

where

0  x  6 

on separate graphs

using rules of amplitude and period. Compare them to y = sin(x) and y = cos(x).

Vertical Translations

- A vertical translation moves the shape of the graph up or down.

- Let

y x c

y x c

cos( )

sin( )

. Then move the graph up/down c units

Examples: Plot y = sin(x) + 2 and y = cos(x) - 3

0  x  2 

on separate graphs using a

table of values. Compare them to y = sin(x) and y = cos(x).

Examples: Plot y = 2sin(3x) - 2 and y = 2cos(2x) - 1 0  x  2 on separate graphs using

rules amplitude, period, and vertical translation. Compare them to y = sin(x)

and y = cos(x).

Reflections about the X-Axis

- Let

cos( )

sin( )

y A x

y A x

. Then

cos( )

sin( )

y A x

y A x

reflects the graphs about the

x-axis

- Recall that sine is odd and cosine is even. We use this to help graph y = sin(-Bx)

and y = cos(-Bx).

Examples: Plot y = -sin(x) and y = -cos(x)

0  x  2 

on separate graphs using a

table of values. Compare them to y = sin(x) and y = cos(x).

Examples: Plot y = - 3sin

x

+ 1 and y = - 2cos

x

- 1 where

0  x  2 

on

separate

graphs using rules amplitude, period, reflection, and vertical translation.

Compare them to y = sin(x) and y = cos(x).

Tangent and Cotangent Graphs

Asymptotes

The graph y =

tan x

has asymptotes at x =

n 

where

n  Z

The graph y =

cot x

has asymptotes at x =

n 

where

n  Z

Examples – Plot the tangent and cotangent graphs using a table of values where

   x  

Amplitude?

- There is no specific amplitude for tangent and cotangent.

Let

cot( )

tan( )

y A x

y A x

. Then if

y A rise fall slower

y A rise fall faster

1 / _

1 / _

Examples – Plot the graphs y = 2tan(x) and y =

cot(x) using a table of values where

Let

csc( )

sec( )

y A x

y A x

. Then if

 

y A closer to x axis

y A further from x axis

1 _ _

1 _ _

Examples – Plot the graphs y = 3sec (x) and y =

2

1

csc(x) using a table of values where

 2   x  2 . Compare to y = sec(x) and y = csc(x)

Period

Let

csc( )

sec

y Bx

y Bx

. Then

B

2 

is the period of the graph.

Examples: Plot y = sec(3x) and y = csc 

x

2

1

where 2   x  2 on separate graphs

using rules of period. Compare them to y = sec(x) and y = csc(x).

Vertical Translations

Let

y x c

y x c

csc( )

sec( )

. Then move the graph up/down c units

Examples: Plot y = sec(2x) + 1 and y = csc

 3 x  +1 where 2   x  2 on separate

graphs using rules of period and translation. Compare them to y = sec(x) and

y = csc(x).

Reflections about the X-Axis

  • Let

csc( )

sec( )

y A x

y A x

. Then

csc( )

sec( )

y A x

y A x

reflects the graphs about the

x-axis

  • Recall that cosecant is odd and secant is even. We use this to help graph

y = sec(-Bx) and y = csc(-Bx).

Examples: Plot y = -sec(x) and y = csc(-2x) where

 2   x  2  on separate graphs

using rules of reflection. Compare them to y = sec(x) and y = csc(x).