VCE Maths Methods - Unit 2 - Circular functions
Circular functions
• Radians & degrees• The unit circle• 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
1
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 2π - 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!
2
π =180°
1 radian = 180°
π
1° = π
180°
VCE Maths Methods - Unit 2 - Circular functions
Radians & degrees
3
Degrees Radians
30°
45°
60°
90°
120°
135°
180°
270°
360°
30π180
=π6
60π180
=π3
90π180
=π2
120π180
=2π3
135π180
=3π4
180π180
= π
270π180
=3π2
360π180
= 2π
45π180
=π4
VCE Maths Methods - Unit 2 - Circular functions
Sin, cos & tan
4
Opposite
Adjacentθ
sin θ( )=
oppositehypotenuse
cos θ( )=adjacent
hypotenuse
tan θ( )=
oppositeadjacent
=sinθcosθ
Hypotenuse
VCE Maths Methods - Unit 2 - Circular functions
Sin, cos & tan - special angles
5
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
6
0°180°
90°
270°
π2
π
3π2
θ
sinθ
cosθ
π4
tan(θ)= sin(θ)
cos(θ)
tanθ
y =sinθ
x =cosθ
1
sin2(θ)+cos2(θ)=1
1.0
1.0
x =cosθ ≈0.71
y =sinθ ≈0.71
tan(θ)= 0.71
0.71=1
0.712+0.712 =1
VCE Maths Methods - Unit 2 - Circular functions
The unit circle - Symmetry
7
0°180°
90°
270°
π2
π
3π2
θ sinθ
cosθ
sin π
2−θ
⎛⎝⎜
⎞⎠⎟=cos θ( )
cos π
2−θ
⎛⎝⎜
⎞⎠⎟=sin θ( )
sinθ
cosθ
π2−θ
sin π
2+θ
⎛⎝⎜
⎞⎠⎟=cos θ( )
cos π
2+θ
⎛⎝⎜
⎞⎠⎟=−sin θ( )
θ
sinθ
cosθ
π2+θ
VCE Maths Methods - Unit 2 - Circular functions
The unit circle - Symmetry
8
0°180°
90°
270°
π2
π
3π2
θ sinθ
cosθ
sin π −θ( )=sin θ( )
sin π+θ( )=−sin θ( )
sin 2π −θ( )=−sin θ( )
cos π −θ( )=−cos θ( )
cos π+θ( )=−cos θ( )
cos 2π −θ( )=cos θ( )
AllSin
Tan Cos
sinθ
cosθθ
π −θ
sinθθ
π+θ
cosθ
sinθθ
2π −θ
cosθ
VCE Maths Methods - Unit 2 - Circular functions
Values of sin x , cos x & tan x
9
AngleAngle Sin θ Cos θ Tan θ
0°
30°
45°
60°
90°
120°
135°
180°
270°
360°
π6π4π3π2
3π4
π
2π
12
112
12 3
2π3
3π2
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
10
sinπ
2=1
sin 3π
2=−1
sinπ =0 sin2π =0
cosπ =−1
cos 3π
2=0
cosπ
2=0
cos0=1
sin0=0
cos2π =1
y = sin xy = cos x
cos x( )=sin x+ π
2⎛⎝⎜
⎞⎠⎟
sin x( )=cos x− π
2⎛⎝⎜
⎞⎠⎟
VCE Maths Methods - Unit 2 - Circular functions
Transformations of sin graphs
11
y = 2sin x
VCE Maths Methods - Unit 2 - Circular functions
Transformations of sin graphs
12
y = sin 2x
VCE Maths Methods - Unit 2 - Circular functions
Transformations of sin graphs
13
y = sin x+2
VCE Maths Methods - Unit 2 - Circular functions
Transformations of sin graphs - summary
14
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 2πb
.
c = vertical translationThe graph is shifted up by c units.
VCE Maths Methods - Unit 2 - Circular functions
Solving circular function equations
15
2sin4x =1
sin4x = 1
2
2sin4x =1 (0<x<π )
4x = π
6(30°)
π4−π
24=
6π24
−π
24
x =5π
24(37.5°)
x = π
24(7.5°)
π2
3π4
π4
π
6π24
12π24
18π24
24π24
5π24
π24
13π24
17π24
π24
+π2=π
24+
12π24
=13π24
(97.5°)
5π24
+π2=
5π24
+12π24
=17π24
(127.5°)
Next two solutions: one period after the "rst two
VCE Maths Methods - Unit 2 - Circular functions
Graphs of tan x
16
Vertical asymptote:
π2
tanπ
4=1
tanθ = sinθ
cosθ As θ → π
2, cosθ → 0, tanθ → ∞
−π2
Period =π
y = tan x
VCE Maths Methods - Unit 2 - Circular functions
Graphs of tan x
17
2tanπ
4=2
y = 2tan x
This graph is dilated from the x-axis by a factor of 2.The period is still the same: π
VCE Maths Methods - Unit 2 - Circular functions
Graphs of tan x
18
tan2π
8=1
y = tan 2x
This graph now has a period of π/2.
π4
−π4
3π4
−3π4
VCE Maths Methods - Unit 2 - Circular functions
Transformations of sin graphs - summary
19
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 πb
.
c = vertical translationThe graph is shifted up by c units.