Transcript Circular Measure Powerpoint Slides

Trigonometry
Radian Measure
Length of Arc
Area of Sector
Area of Segment
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

s=
 2r
2
s = 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
r
θ
Length of arc = r
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
= 0·628 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 = 2.5 θ
θ = 900
2·5
θ = 360 radians
Area of Segment
ASegment = AS ec tor  ATriangle
ASegment
Q5.
1 2
1 2
= r   r sin 
2
2
Find the area of the shaded region.
Find the area of the shaded region.
4
 = 2 cos  
6
 = 1.682137341
1
ASegment = AS ec tor  ATriangle
ASegment
ASegment
1 2
1 2
= 6 (1.682137341)  6 sin(1.682137341)
2
2
2
= 12.4units