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VCE Maths Methods - Unit 2 - Circular functions
Circular functions
• Radians & degrees• Sin, cos & tan• The unit circle• Values of sin x , cos x & tan x• Graphs of sin x & cos x• Transformations of sin graphs• Solving circular function equations• Graphs of tan x
2
VCE Maths Methods - Unit 2 - Circular functions
VCE Maths Methods - Unit 1 - Circular functions 4
0°180°
90°
270°
!
2
!
3!2
!
4
Radians & degrees
• The radian is another measure of angles.
• A circle with a radius of 1 has a circumference of 2! - this is the basis of the radian measure.
• It is very useful as it is a number with no unit.
• Make sure that your calculator is in radians mode & know how to change it!
3
! =180°
1 radian = 180°
!
1° = !
180°
VCE Maths Methods - Unit 2 - Circular functions
Radians & degrees
4
Degrees Radians
30°
45°
60°
90°
120°
135°
180°
270°
360°
30!180
=!6
60!180
=!3
90!180
=!2
120!180
=2!3
135!180
=3!4
180!180
= !
270!180
=3!2
360!180
= 2!
45!180
=!4
VCE Maths Methods - Unit 2 - Circular functions
Sin, cos & tan
5
Opposite
Adjacent!
sin !( )=
oppositehypotenuse
cos !( )=adjacent
hypotenuse
tan !( )=
oppositeadjacent
=sin!cos!
Hypotenuse
VCE Maths Methods - Unit 2 - Circular functions
Sin, cos & tan - special angles
6
90°
1
45°
45°
1
2
sin 45°( )=
oppositehypotenuse
=12
cos 45°( )=
adjacenthypotenuse
=12
tan 45°( )=
oppositeadjacent
=11=1
1 1
2 2
60°
30°
sin 30°( )=
12
cos 30°( )=
32
tan 30°( )=
13
3
sin 60°( )=
32
cos 60°( )=
12
tan 60°( )=
31= 3
VCE Maths Methods - Unit 2 - Circular functions
The unit circle
7
0°180°
90°
270°
!2
!
3!2
!
sin!
cos!
!4
tan(!)= sin(!)
cos(!)
tan!
y =sin!
x =cos!
1
sin2(!)+cos2(!)=1
1.0
1.0
x =cos! "0.71
y =sin! "0.71
tan(!)= 0.71
0.71=1
0.712+0.712 =1
VCE Maths Methods - Unit 2 - Circular functions
The unit circle - Symmetry
8
0°180°
90°
270°
!2
!
3!2
! sin!
cos!
sin !
2"#
$%&
'()=cos #( )
cos !
2"#
$%&
'()=sin #( )
sin!
cos!
!2"#
sin !
2+"
#$%
&'(=cos "( )
cos !
2+"
#$%
&'(=)sin "( )
!
sin!
cos!
!2+"
VCE Maths Methods - Unit 2 - Circular functions
The unit circle - Symmetry
9
0°180°
90°
270°
!2
!
3!2
! sin!
cos!
sin ! "#( )=sin #( )
sin !+"( )=#sin "( )
sin 2! "#( )="sin #( )
cos ! "#( )="cos #( )
cos !+"( )=#cos "( )
cos 2! "#( )=cos #( )
AllSin
Tan Cos
sin!
cos!!
! "#
sin!!
!+"
cos!
sin!!
2! "#
cos!
VCE Maths Methods - Unit 2 - Circular functions
Values of sin x , cos x & tan x
10
AngleAngle Sin θ Cos θ Tan θ0°
30°
45°
60°
90°
120°
135°
180°
270°
360°
!6!4!3!2
3!4
!
2!
12
112
12 3
2!3
3!2
32
0 0 1 0
32
13
12
1 0 !!!
32
!12 ! 3
!112
!12
0 !1 0
!1 0 !!!
0 1 0
VCE Maths Methods - Unit 2 - Circular functions
Graphs of sin x & cos x
11
sin!
2=1
sin 3!
2="1
sin! =0 sin2! =0
cos! ="1
cos 3!
2=0
cos!
2=0
cos0=1
sin0=0
cos2! =1
y = sin xy = cos x
cos x( )=sin x+ !
2"#$
%&'
sin x( )=cos x! "
2#$%
&'(
VCE Maths Methods - Unit 2 - Circular functions
Transformations of sin graphs - summary
15
y = asinbx+c
a = amplitude of graphThe height that the graph goes
above the midpoint.
b = period factor
The period of the function is found from 2!b
.
c = vertical translationThe graph is shifted up by c units.
VCE Maths Methods - Unit 2 - Circular functions
Solving circular function equations
16
2sin4x =1
sin4x = 1
2
2sin4x =1 (0<x<! )
4x = !
6(30°)
!4"!
24=
6!24
"!
24
x =5!
24(37.5°)
x = !
24(7.5°)
!2
3!4
!4
!
6!24
12!24
18!24
24!24
5!24
!24
13!24
17!24
!24
+!2=!
24+
12!24
=13!24
(97.5°)
5!24
+!2=
5!24
+12!24
=17!24
(127.5°)
Next two solutions: one period after the "rst two
VCE Maths Methods - Unit 2 - Circular functions
Graphs of tan x
17
Vertical asymptote:
!2
tan!
4=1
tan! = sin!
cos! As ! " #
2, cos! " 0, tan! " $
!"2
Period =!
y = tan x
VCE Maths Methods - Unit 2 - Circular functions
Graphs of tan x
18
2tan!
4=2
y = 2tan x
This graph is dilated from the x-axis by a factor of 2.The period is still the same: !
VCE Maths Methods - Unit 2 - Circular functions
Graphs of tan x
19
tan2!
8=1
y = tan 2x
This graph now has a period of !/2.
!4
!"4
3!4
!3"4
VCE Maths Methods - Unit 2 - Circular functions
Transformations of tan graphs - summary
20
y = atanbx+c
a = dilation factor of grapha >1,the graph is steeper.
0 < a< 1,the graph is less steep.
b = period factor
The period of the tan function is found from !b
.
c = vertical translationThe graph is shifted up by c units.