Circular Measures

WAJA 2009

ADDITIONAL MATHEMATICS FORM FOUR

( Student’s Copy )

Name: ___________________________ Class : ___________________________

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Activity Test yourself: How much do you know? Match the diagrams on the left to the appropriate term on the right. (O is the centre of the circle)

minor sector

minor arc

major sector

O

O

chord angle subtended by arc at the centre of a circle

O

major arc

segment

tangent

O

2

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Learning Objective: 8.1 – Understand the concept of radian Learning Outcomes: 8.1.1 – Convert the measurements in radians to degrees and vice versa. A. Understanding the concept of Radian Observe the following diagram. A'

r

r r



unit

O

A

O

A

r

1 radian Circle with centre O and radius r unit.

OA is rotated about O such that the length of arc AA’ is equal to the radius of the circle.

Hence, fill in the blanks with the correct words given. 52

radian

arc

radius

centre

AOA’ is an angle subtended at the …………………. of the circle by an ………… equal in length to the ………………… of the circle. The measure of this angle is defined as one ……………….. Radian is just a new unit introduced to describe the magnitude of an angle! Measure the angle AOA’ with a protractor. 1 rad. = ……………. (round to the nearest degree.) 1. Find the angle subtended by arc in radian.

(a)

Diagram

Radius

Length of arc

A’

r

r

Angle subtended at the centre, θ (in radian)

1

3

rad

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Diagram

Length of arc

r

1.5 r

Angle subtended at the centre, θ (in radian)

r

r O

(b).

Radius

A

r

A’ 1.5 r

r

A

r

1.5

(c)

rad

2r r r

r

2r rad

(c)

3r

r

r

r

3r rad

r

(d) 180

r

r

r

r 180 =

rad

(answer in term of ) (e)

r 4

6r

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Diagram

Radius

Length of arc

Angle subtended at the centre, θ (in radian)

r

6r

rad

(f).

OA makes one complete revolution, i.e OA has moved through an angle of 360 2 r

r

r

2 r

(g)

(answer in term of )

360 =

rad

If r = 1.6 cm and s = 4 cm. 4 cm



r = 2 cm

2 cm

(h)

4 cm

r cm

=

rad

If r = 2.0 cm and s =3.5 cm. 3.5 cm r = 2 cm 2 cm

Conclusion: 5

3.5 cm

r cm

=

rad

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

I. In general, the measure of angle (in radian) subtended by arc of length s and radius r is given by s



r

θ (in rad) = 



r

O

II From question 1(d) and 1(f) above, relationship between measure of angle in degree and radian is given by 2 rad =………  rad =………

1 rad 

1 

180

 180



 ………… 

rad  ………………….. rad (4 s.f.)

2. Convert the following angles from degrees to radians. (Give your answers in terms of ). (a) 15

(b) 30

Solution: 1 

 180

=> 15 = 15  =

 12

(c) 90

Solution: 1 

rad

 180

rad

rad

 180

Solution: 1 

rad

=> 30 = 30 

rad

=

rad

6

 180

=> 90 = 90  =

rad rad rad

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

(d) 45

(e) 315

(f) 300

3. Convert the following angles from radians to degrees. (Give your answers correct to two decimal places where necessary) (a) 0.5 rad

(b)

Solution: 1 rad 

180



=> 0.5 rad = 0.5 

2 3

 rad

Solution: 1 rad 

180



= 28.65

2 => 3 180



(c)

2 rad = 3



(d) 4.562 rad

(e)

 rad

Solution:

180



5 6

1 rad 

 

5 => 6

180



 rad =

5 6



=

 5

= 120 (f) (2  1.5) rad

rad

4. Convert the following angles from degrees to radians. Give your answers correct to 4 significant figures: 7

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

(a) 20

(b) 50

Solution: 1 

Solution:

 180

=> 20 = 20 

(c) 100

1 

rad

 180

rad

 180

Solution: 1 

rad

=> 50 = 50 

 180

rad

rad

=> 100 = 100 

rad

=

rad

= 0.3491 rad =

(d) 230

(e) 70.4

rad

(f) 30830’ [Remember 1 = 60’]

5. A table of radians and degrees for some special angles is shown below. Complete the blanks. Angles in degrees Angles in radians

0

30

60





5

4

90

180



8

3 2

2

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Learning Objective: 8.2 Understand and use the concept of length of arc of a circle to solve problems. Learning Outcomes: 8.2.1 (a) Determine length of arc B r O

s

Figure shows a sector OAB, with centre O, radius r, arc length AB = s cm and angle AOB =  radians.

 r

A

s Based on the definition the radian, i.e.   , derive a formula for finding the length of the r arc.

Calculate the length of arc AB for each of the following circles or sectors. Diagram 1.

radius r

 (in rad)

Arc length s

s 7 cm B

2.73 rad O 7 cm

4.412 rad O 5 cm

= 19.11 cm

B

A

9 B

s = rθ = 7  2.73

A

2. s

2.73 rad

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Diagram 3.

B

radius r

 (in rad)

s

1.95 rad 8 cm

O 4.

A

B s

1.45 rad 18 cm

O

A

5.

10 cm s

1 rad

A

B 6. 26 =

s 26

rad

18 cm

7. Major arc AB

360…….

A

=………. 100 B

9.5 cm

= ……..rad O

`

10

Arc length s

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Learning Objective: 8.3 – Understand and use the concept of area of a sector of a circle to solve problems. Learning Outcomes: 8.3.1(a) Determine the area of a sector Consider a circle of center O and radius 1.36 cm. OB is rotated to form a sector BOB’. B'

O

B'

60 r

120 

B

O

B

r

240 

180 

B'

O

r

B

O

r

B

….

B'

Angle subtended by arc BB’ Area of sector/cm2

0

60

120

180

240

300

0

0.97

1.94

2.91

3.87

4.84

Area of sector/ cm2 5

4

3

2

1

50

100

150

200

250

300

angle/

Observe the above diagram. What happen to the area of sector as the angle  increases? …………………………………………………………………………………………… The area of sector is …………………………………. to the angle subtended at the centre of the circle.

11

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Area of sector BOB' Angle subtended by arc BB' in radian  Area of circle Angle subtended by circle in radian

Let area of sector BOB’ be A. Derive the formula for area of sector, A in term of r and  based on the above relationship.

ICT: Open Microsoft PowerPoint Slide Show in WAJA CD to view a pictorial representation of the formula of area of a sector. Q

1. The diagram shows a circle with centre O and diameter PR. Given that PR=20 cm, and POQ = 1.2 rad. Find (a) the area of minor sector POQ (b) the area of minor sector QOR

1.2 rad.

O

(a)

R 1 2 r  2 1  (10) 2 (1.2) 2 = 60 cm2

A

r= ½ PR rad. = ½ (20) = 10 cm

 = 1.2

(b) r=

2. Calculate the area of sectors for each of the following diagram.

12

=

P

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Diagram

radius r

 (in rad)

(a) 5.5 cm 0.62 rad

(b) Major sector ABC

A

s

3.3 rad B 5 cm C

(c) Sector JOK K

0.82 rad

J

O Convert angle to radian.

(d)

136

13

Area of sector A

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Diagram

radius r

(e) Major sector HOG

O.

G. 1.8 rad.

H.

14

 (in rad)

Area of sector A

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 4) Chapter 8: Circular Measure

Activity A r



r

B

O For a sector OAB, with centre O, radius r, arc length AB=s cm and AOB= radians, find the missing values in the table below: [ use formula s=r and A  r cm

 radians

(a)

4

1.25

(b)

6

6

(d)

0.8

(e)

1.2

(f)

s cm

(g)

9.6 60 64 6

5.5

(c) s = r 14 = r r= = 26.74 cm ………………………… Area of sector = = 187.17 cm2

14

8

(h)

Area of sector / cm2

9



(c)

1 2 r  ] 2

27 30.25

Arc length, s = r =5.5  2 = 11 cm

(h)

  = 2 rad.

15