Trigonometry 2

1
Trigonometry (2)

• Radians
• Area
• Arcs
• Area of triangle

2
Radians (1)
y
90
1
x
0
180
270
A unit circle has radius 1. The circumference is
2?r 2 ?
3
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
4
Radians (3)
0o 0 radians
90o ?/2 radians
180o ? radians
270o 3 ?/2 radians
360o 2? radians
Example
Example
5
Why Radians (1)
?/2
Its very handy for working out the lengths of
arcs
arc a
r
?
0
?
2?
3?/2
Example
6
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
7
Why Radians (3)
?/2
Its very handy for working out the area of
sectors
r
sector
?
0
?
2?
3?/2
Example
8
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
9
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
10
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
11
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