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UNIT 3: PURE MATHEMATICS 1 – CIRCULAR MEASURE
CIRCULAR MEASURE
P
r
Candidates should able to:



Understand and define angles in radian;
Find the arc length and area of sector/segment;
Use the formulae 𝑠 = 𝑟𝜃 and 𝐴 =
1 2
𝑟 𝜃
2
𝜃
r
Q
Definition of 1 radian
If arc length PQ = radius r,
then angle POQ, 𝜃 = 1
radian
O
in solving problems concerning the arc
length and sector area of a circle.
1.
Units for Measuring Angles
The most common unit for measuring an angle is degree. The degree system is based on
the division of one complete revolution into an arbitrary choice of 360 equal angles. We
define the size of each angle to be one degree or 1°.
There is also the gradient system in which one complete revolution is divided into 400
equal angles. The size of each angle in this case is defined to be one gradient or 1 gra.
Note:
Regardless of the size of the circle, 𝜃 =
1 radian as long the length of the arc is
equal to the radius of the circle.
Notice that the above two systems are based on an arbitrary choice of division. There is a
system that does not depend on the choice of any particular number. It is the radian
system. This system of angular measurement, known as circular measure, is applied more
frequently in mathematics involving Calculus.
The radian is used much more than the degree in higher mathematics for measuring
angles.
1.1 Radian Measure
s
In general, if the length of the arc is s units and the
𝑠
radius is r units, then 𝜃 = . That is, the size of the
𝜃
𝑟
r
angle is given by the ratio of the arc length to the
radius.
𝜽(𝒊𝒏 𝒓𝒂𝒅) =
A radian is the angle subtended at the centre of a circle by an arc length whose length is
equal to that of the radius of the circle. This means a radian formed when the arc length
and the radius are the same.
𝒔
𝒓
1
UNIT 3: PURE MATHEMATICS 1 – CIRCULAR MEASURE
1.2 Relation between Radian and Degree measure
1.2.2 Changing Radians to Degrees
Consider the angle 𝜃 in a semicircle of radius r as shown below.
𝜃=
e.g 4: Convert
𝑠
s
𝑟
𝑠 = 𝑟𝜃
Since 𝑠 =
1
2
𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒, so we get
𝜃
1
𝑠 = (2𝜋𝑟) = 𝜋𝑟.
2
r
Then, 𝑟𝜃 = 𝜋𝑟, we get 𝜃 = 𝜋 radians.
In a semicircle, 𝜃 = 180°, hence we get
𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 = 𝟏𝟖𝟎°
2𝜋
3
radians to degrees.
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 180°
180
1 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
𝜋
2𝜋
2𝜋 180
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
×
3
3
𝜋
= 120°
e.g 5: Convert 2.1 radians to degrees.
180
2.1 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 2.1 ×
𝜋
= 120.3°
1.2.1 Changing Degrees to Radians
Since 180° = 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠, so 1° =
𝜋
180
Leave your answer in
e.g 6: Convert the following angles from radians to degrees.
𝜋
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 90°
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
2
6
𝜋
3𝜋
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
°
4
2
𝜋
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
°
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
3
terms of  unless asked
for more accuracy
Homework:
Ex 18A Pg 267 Qn 1 and Qn 2
.
e.g 1: Convert 45° to radians
1° =
𝜋
180
𝜋
45° = 45 ×
180
𝜋
=
4
e.g 2: Convert 75° to radians, give your answer to 3 s.f.
𝜋
1° =
180
𝜋
75° = 75 ×
180
= 1.308996 = 1.31 (3 𝑠. 𝑓. )
e.g 3: Convert the following angles to degrees.
30° =
45° =
81.5° =
2.
°
°
°
Arc Length and Area of Sector
2.1 Length of an Arc
The length of an arc is always proportional to the angle at the center of the arc and the
radius of the arc. If 2 arcs have the same radius but one has an angle twice the size of the
other, it means on arc length will be twice the size of the other.
48° =
60.7° =
333° =
l
2l
2
UNIT 3: PURE MATHEMATICS 1 – CIRCULAR MEASURE
The arc length 𝑠 is directly proportional to the angle 𝜃
subtended at the center of the circle. We therefore have:
Now, as 𝑠 = 𝑟𝜃, we have:
s
1
1
𝐴 = 𝑟(𝑟𝜃) = 𝑟𝑠
2
2
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑠
𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑏𝑡𝑒𝑛𝑑𝑒𝑑 𝑏𝑦 𝑠 𝑎𝑡 𝑂
=
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
𝑡𝑜𝑡𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑒 𝑂
The area, A, of a sector is given by:
𝜃
𝑠
𝜃
=
2𝜋𝑟 2𝜋
1
1
𝐴 = 𝑟 2 𝜃 𝑜𝑟 𝐴 = 𝑟𝑠
2
2
where 𝜃 is in radians.
r
So Length of arc, 𝒔 = 𝒓𝜽
where 𝜃 is in radians.
e.g 7: Find the length of the arc ABC.
𝑠 =5×
=
10𝜋
3
Important: If the question gives the angle at the centre in degrees, it must be changed to
radians in order to apply the area of sector formula.
e.g 8: Find the radius of the sector ABC.
e.g 10: Find the area of sector ABC, where
𝜋
∠𝐴𝐵𝐶 = and r = 2cm, give your answer in
3
terms of 𝜋.
2𝜋
3
cm or 10. 5cm
e.g 11: Find the area of sector ABC, where
∠𝐴𝐵𝐶 = 60° and r = 8cm, give your answer
to 3 s.f.
e.g 9: In a circle of radius 8 cm, find
a) the length of the arc which subtends an angle of
3𝜋
4
Homework:
Ex 18A Pg 268 Qn 4
radians at the centre,
b) the angle subtended by arc of length 6cm.
3.
2.2 Area of a sector of a circle
In the diagram, the angle of sector is 𝜃 radians.
s
In degree:
area of a sector, 𝐴 =
𝜃
× 𝜋𝑟 2
360°
In radian: (𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 180°)
𝜃
× (180°)𝑟 2
360°
1
𝐴 = 0.5𝜃𝑟 2 = 𝑟 2 𝜃
2
A
𝜃
r
Area of a Segment
While a sector looks like a “pizza” slice, a segment looks like
the “pizza” slice with the triangular portion cut off. The
segment is only the small partially curved figure left when the
triangle is removed.
Dealing with the area of a segment is very similar to working
with area of a sector. If you find the area of the sector and
subtract the area of the triangle, you will have the area of the
segment portion of the circle.
Therefore, 𝐴 =
𝑨𝒔𝒆𝒈𝒎𝒆𝒏𝒕 = 𝑨𝒔𝒆𝒄𝒕𝒐𝒓 − 𝑨𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆
3
UNIT 3: PURE MATHEMATICS 1 – CIRCULAR MEASURE
3.1 Area of a Triangle
(a) the area of the shaded region = (6𝑟 + 9) 𝑐𝑚2 .
The most common formula for the area of a triangle would be:
(b) the perimeter of the shaded region equals to 6 + 𝜃(2𝑟 + 3)𝑐𝑚.
(c) Given that the shaded region is three quarters of the area of the sector 𝑂𝐴𝐵, calculate
the value of r.
(d) If, however, the total perimeter of the shaded region equals the total perimeter of the
sector 𝑂𝐴𝐵, find the value of 𝜃.
𝐴𝑟𝑒𝑎 =
𝜃
2
1
× 𝑏𝑎𝑠𝑒(𝑏) × ℎ𝑒𝑖𝑔ℎ𝑡(ℎ)
2
But not all questions provide the right information to use this formula. Another formula
that can be used to obtain the area of a triangle uses the sine function. It allows us to find
the area of a triangle when we know the lengths of two sides and the size of angle
between them.
By using the right triangle on the left side of the
diagram, and our knowledge of trigonometry, we
cab state that :
ℎ
sin 𝐶 =
𝑏
𝑏 sin 𝐶 = ℎ
This tells us that the height, h, can be expressed as
𝒃 𝐬𝐢𝐧 𝑪.
If we substitute this expression for the height in the
original area of triangle formula, we can write:
𝟏
𝑨𝒓𝒆𝒂𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆 = 𝒂𝒃 𝐬𝐢𝐧 𝑪
𝟐
Alternate version are:
1
𝐴𝑟𝑒𝑎 = 𝑎𝑐 sin 𝐵
2
1
𝐴𝑟𝑒𝑎 = 𝑏𝑐 sin 𝐴
2
or
e.g 15: The figure below shows a circle, centre O, radius 5 cm and two tangent TA and TB,
each of length 8 cm. Calculate
(a) ∠𝐴𝑂𝐵,
(b) the length of arc 𝐴𝑃𝐵,
(c) the area of the shaded region.
Homework:
Ex 18A Pg 268 Qn 5, 6, 7
4.
Brief recap of Secondary level trigonometry
4.1 Trigonometric Ratios of Acute Angles
The following formulae link the sides and angles in right-angled triangles:
𝜋
e.g 12: In the figure, O is the centre of a circle of radius 6 cm and ∠𝐴𝑂𝐵 = . Find
3
𝑂
𝐻
𝐴
cos 𝑥 =
𝐻
𝑂
tan 𝑥 =
𝐴
where H is the length of the hypotenuse;
O is the length of the side opposite the angle;
A is the length of the side adjacent to the angle.
sin 𝑥 =
(a) the area of sector 𝐴𝑂𝐵,
(b) the area of ∆𝐴𝑂𝐵.
Hence, find the area of the shaded segment.
e.g 13: In the figure, O is the centre of a circle of radius 8 cm and ∠𝐴𝑂𝐵 = 1.8 𝑟𝑎𝑑. Find
(a) the area of the shaded region,
(b) the perimeter of the shaded region.
H
O
x
A
These formulae are often remembered using the acronym
SOHCAHTOA or by using this : Silly Old Harry Couldn’t
Answer His Test On Algebra.
e.g 14: In the figure, O is the centre of the circle containing the sector 𝑂𝐴𝐵. 𝐴𝐵 and 𝐷𝐶 are
arcs of two concentric circles. 𝐵𝐶 = 3𝑐𝑚.
If 𝑂𝐶 = 𝑟 𝑐𝑚and ∠𝐴𝑂𝐵 = 𝜃 𝑟𝑎𝑑, show that
4
UNIT 3: PURE MATHEMATICS 1 – CIRCULAR MEASURE
4.2 Sine Rule
5.
The sine rule connects the length of sides and angles in any triangle 𝐴𝐵𝐶.
It states that:
𝑎
𝑏
𝑐
C
=
=
sin 𝐴 sin 𝐵 sin 𝐶
Radians and degrees are two different units that your calculator will use to measure angles.
When you enter the information about an angle into your scientific calculator, it will give
you an answer either in units of radians or units of degrees.
An alternate version of the formula is:
sin 𝐴
𝑎
=
sin 𝐵
𝑏
=
For instance, if you want to calculate sin 30°, you must put your calculate in degree mode.
a
b
𝜋
Similarly, if you want the answer for cos( ), make sure your calculator is in radian mode
4
sin 𝐶
𝑐
Degree and Radian mode in calculator
A
c
before solving for the answer.
B
Important: We can use the sine rule when we are given –
 Two sides and an angle opposite to one of the two sides.
 One side and any two angles.
Pressing “mode” or “shiftmode”, should able to guide you to the mode setting screen in
your calculator.
4.3 Cosine Rule
6.
The cosine rule also connects the length of sides and angles in any triangle 𝐴𝐵𝐶.
Find the exact values of Trigonometric ratios. What this means is don’t use your calculator
to find the value (which will normally be a decimal approximation – unless you have the
latest model calculators). The trigonometric ratios of angles measuring 30°, 45° and 60°
can be obtained using right angles triangles formed by taking “half” of a square and an
equilateral triangle.
It states that
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
Equivalently, we also have these formulae:
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
C
a
b
A
c
Trigonometric Ratios of Some Special Angles
B
Important: We can use Cosine rule when we are given –
 Two sides and the included angle.
 Three sides.
e.g 16
Trigonometric types
30° 𝑜𝑟
𝜋
6
Angles, 𝜃
𝜋
4
45° 𝑜𝑟
60° 𝑜𝑟
𝜋
3
cos 𝜃
sin 𝜃
tan 𝜃
5