Trigonometry

P1^

7

The following formulae often come in useful when solving problems involving

sectors of circles.

For any triangle ABC:

The sine rule: (^) sinsaA= inbB=sincC
or sinsA in sin
a

B

b

C

c

= =

The cosine rule: a^2 = b^2 + c^2 − 2 bc cos A

or cosA=bc+−bc a

22 2

2

The area of any triangle ABC = 12 ab sin C.

ExAmPlE 7.12 Figure 7.31 shows a sector of a circle, centre O, radius 6 cm. Angle AOB = 23 π

radians.

(i) (a) Calculate the arc length, perimeter

and area of the sector.

(b) Find the area

of the blue

region.

(ii) Find the exact length of

the chord AB.

SOlUTION

(i) (a) Arc length = rθ

=×^6

2

3

π

= 4 π cm

Perimeter = 4 π + 6 + 6 = 4 π + 12 cm

Area = 21 r^2 θ =× 21 62 ×^23 π = 12 π cm^2

(b) Area of segment = area of sector AOB – area of triangle AOB

The area of any triangle ABC = 12 ab sin C.

Area of triangle AOB =×^1 ×= =

2

66 2

3

18 3

2

sinπ 93 cm^2

(^) So area of segment =−

12 93

221

π

.cm^2

%

$

b

F

D

&

Figure 7.30

6 cm 6 cm

O

A B

2 π

3

Figure 7.31

This is called a

segment of the circle.