Transcript Trig1 - resources.teachnet.ie

Trigonometry
Radian Measure
Length of Arc
Area of Sector
Radian Measure

To talk about trigonometric functions, it is
helpful to move to a different system of
angle measure, called radian measure.

A radian is the measure of a central angle
whose minor arc is equal in length to the
radius of the circle.

There are 2 or approximately 6.28318,
radians in a complete circle. Thus, one
radian is about 57.296 angular degrees.
Radian Measure
r
1 radian
r
Radian Measure



There are 2π radians in a full rotation –
once around the circle
There are 360° in a full rotation
2π = 360°
π = 180°
To convert from degrees to radians or
radians to degrees, use the proportion
degrees radians
=
180 o

Examples
 Find the degree
measure equivalent
of 3π radians.
4
3π 3180
=
4
4
= 135°
 Find the radian measure
equivalent of 210°.
180° = π
π
1° =
180
210π 7π
210° =
=
180
6
Length of Arc
Fraction of circle
θ must be
in radians
r
θ
Circumference = 2πr
l

2
Length of arc

l=
 2 r
2
l = r
Area of Sector
Fraction of circle
θ must be
in radians

2
r
θ
Area of circle = π r
Area of sector
2

Area =
 r2
2
1 2
= r
2
1 2
Area of sector = r 
2
θ must be
in radians
Page 7 of tables
r
θ
Length of arc = r
Page 6 of tables
Examples

A circle has radius length 8 cm. An angle of 2.5
radians is subtended by an arc.
Find the length of the arc.
l = rθ
l = 2·5  8
= 20 cm
l
2·5
8 cm
(i)
Find the length of the minor arc pq.
(ii)
Find the area of the minor sector opq.
Q1.
Q2.
p
p
5
6
10 cm
o
0·8 rad
q
l = rθ = 10(0·8) = 8 cm
1 2
1
A = r  = (10) 2 (0·8)
2
2
= 40 cm2
12 cm
o
q
5
= 10 cm
l = rθ = (12)
6
1
1 2
2 5
A = r  = (12)
2
6
2
= 60 cm2
Q3.
The bend on a running track is a semi-circle of radius
100 metres.
π
A
20 m
A runner, on the track, runs a distance of 20 metres on
the bend. The angles through which the runner has run is A.
Find to three significant figures, the measure of A in radians.
l = rθ
100
20 =
θ
π
π
θ = 20 
100
=
0·6283..
= 0·628 radians
Q4.
A bicycle chain passes around two circular cogged wheels.
Their radii are 9 cm and 2·5 cm. If the larger wheel turns
through 100 radians, through how many radians will the
smaller one turn?
100 radians
2·5
9
l = rθ
l = 9  100 = 900 cm
900 =
2·5θ 900
θ=
2·5
θ = 360 radians
The diagram shows a sector (solid line) circumscribed by a circle
(dashed line).
(i) Find the radius of the circle in terms of k.
cos 30º =
k
2
3
cos 30º =
r
2
k
3
2
=
r
2
3
k
=
r
1
r=
k
3
2006 Paper 2 Q4 (c)
k 60º
2 30º
r
k
r
k
The diagram shows a sector (solid line) circumscribed by a circle
(dashed line).
(ii) Show that the circle encloses an area which
is double that of the sector.
Area of circle = π r2
r=
k
3
 k 2
=π 
 3
k2
Twice area of sector
=
3
1 2
1 2    k2
Area of sector = r  = k   =
2
2 3
6
2006 Paper 2 Q4 (c)

3
r
k
r
k