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Trigonometry (2)

- Radians
- Area
- Arcs
- Area of triangle

Radians (1)

y

90

1

x

0

180

270

A unit circle has radius 1. The circumference is

2?r 2 ?

Radians (2)

y

As well as degrees, angles can be expressed in

RADIANS

?/2

1

?

0

x

RADIANS are the distance traveled around the unit

circle.

?

2?

3?/2

0o 0 radians

180o ? radians

360o 2? radians

90o ?/2 radians

270o 3 ?/2 radians

Radians (3)

0o 0 radians

90o ?/2 radians

180o ? radians

270o 3 ?/2 radians

360o 2? radians

Example

Example

Why Radians (1)

?/2

Its very handy for working out the lengths of

arcs

arc a

r

?

0

?

2?

3?/2

Example

Why Radians (2)

?/2

1 radian is defined as- the angle subtended at

the centre of a circle radius r by an arc of

length r

r

r

1c

0

2?

?

A c superscript is used to denote radians

3?/2

Why Radians (3)

?/2

Its very handy for working out the area of

sectors

r

sector

?

0

?

2?

3?/2

Example

Length arc r? 15 x 2 30 cm

Area unshaded 1/2r?2 0.5 x 2 x 152 225

Area circle ?r2 ? x 152 225 ?

Area shaded 225? - 225

Formula Book

Area of triangle 1/2 base x height 1/2

b x h

B

sin C Opp/Hyp h/a

a

c

h a sin C

h

C

A

b

Area 1/2 b x a sin C

The area proof - wont be examined

Area 1/2 ab sin C

The Area of a Triangle

for working the area in non-right angled

triangles

Area 1/2 ab sin C

C

angles

a

b

sides

B

A

c

Area 1/2 ac sin B

or

Area 1/2 bc sin A

or

Area - example

Area 1/2 ab sin C

C

angles

sides

75o

7 cm

4 cm

B

A

Area 1/2 ab sin C

Area 1/2 x 4 x 7 sin 75

Area 14 x 0.966

Area 13.5 cm2 1 d.p.

Example