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Ng Wei Shen
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Saturday, February 02, 2008
Main Important Formulas

1.Length of Arc = r ө (where r is radius and ө is in radians)
2.Area of Sector = ½ r² ө
= ½ r S (where rө= S)
3.Area of Segment = ½ r² (ө - sin ө)
4.∏ radians = 180°

Supplementary Excercise
Topic: Circular Measure

1. 1999/S3/T4/Q5
(i) Area of Segment = ½ r² (ө - sin ө)
= ½ (5)² (0.825 – sin 0.825)
= 1.130
= 1.13 (3 s.f.)

(ii) Angle APO = (∏ - 0.825) ∕ 2
= 1.1583
Area of Sector APB = ½ r² ө
= ½ (4)² (2∏– 1.1583 – 1.1583)
= 31.7327
Area of Shaded Region = 31.7327 – 1.130 – 1.130
= 29.47
= 29.5 (3 s.f.)

2. 2000/S3/T4/Q5
(i) cos BAC = 9 / 10
Angle BAC = cos¯¹ ( 9/10 )
= 0.4510
= 0.451 rad (3 s.f.)

(ii) BC² + 9² = 10²
BC = √(100 – 81)
= 4.3589
Area of ABC = ½ (4.3589) (9)
= 19.615
Area of ABD = ½ (9²) (0.451)
= 18.2655
Area of Shaded Region = 19.615 – 18.2655
= 1.3495
= 1.35 cm²
(iii) Angle BOC = 0.451 + 0.451
= 0.902
Length of Arc BC = (5) (0.902)
= 4.51

3. 2002/S3/T6/Q4
(i) Length PB = ½ (6) (∏ / 3)
= 3.1416
Length PQ = (2) ( 2∏ / 3)
= 4.1888
OQ² + 2² = 4²
OQ = √( 16 – 4)
= 3.4641
Perimeter of Shaded Region = 3.1416 + 4.1888 + (6 – 3.4641)
= 9.8663
= 9.87 ( 3 s.f.)

(ii) Area of OBP = ½ (6²) (∏ / 6)
= 9.4248
Area of COP = ½ (2)² (2∏ / 3)
= 4.1888
Area of OQC = ½ (2) (3.4641)
= 3.4641
Area of Shaded Region = 9.4248 – 4.1888 – 3.4641
= 1.7719
= 1.77 cm² (3 s.f.)

4. 2005/S4/T1/Q3
(a) cos TCB = 10 / 30
TCB = cos¯¹ (10/30)
= 1.23096

(b) Length = (10) (∏ - 1.23096)
= 19.106
= 19.1 cm

(c) Angle TBC = ∏ / 2 – 1.23096
0.339836
Angle ODB = 0.339836 (isosceles)
Andle DOA = 0.339836 + 0.339836
= 0.6797 (4 s.f.)


5. 2006/S4/T1/Q1
(a) Area = (0.5)(8²)(1.2-Sin1.2)
= 8.57cm²

(b) Length = (∏)(16)-(1.2)(8)
= 40.7cm

(c) Radius = (∏)(16)-(1.2)(8)(0.5)/(∏)
= 6.47cm

6. 2006/S4/T1/Q5
(a) cos PBC = 7 / 25
PBC = cos¯¹ ( 7 / 25)
= 1.287
= 1.29 (3 s.f.)

(b) Angle PAD = ∏ / 2 + (∏/2 – 1.287)
= 1.8546
DP² = 9² + 9² - 2 (9) (9) (cos 1.8546)
= 207.36
DP = 14.40