circular functions · vce maths methods - unit 2 - circular functions circular functions •...

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 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°


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


Sin, cos & tan

5

Opposite

Adjacent!

sin !( )=

oppositehypotenuse

cos !( )=adjacent

hypotenuse

tan !( )=

oppositeadjacent

=sin!cos!

Hypotenuse


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


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


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+"


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!


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


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#$%

&'(


Transformations of sin graphs

12

y = 2sin x



13

y = sin 2x



14

y = sin x+2


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.


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


Graphs of tan x

17

Vertical asymptote:

!2

tan!

4=1

tan! = sin!

cos! As ! " #

2, cos! " 0, tan! " $

!"2

Period =!

y = tan x


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: !


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


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.

circular functions · vce maths methods - unit 2 - circular functions circular functions •...

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