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.