UNIT 3: PURE MATHEMATICS 1 CIRCULAR MEASURE

CIRCULAR MEASURE

Candidates should able to:

1. Understand and define angles in radian;

1. Find the arc length and area of sector/segment;

1. Use the formulae and 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 .

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.

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

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.

(P)

(Definition of 1 radianIf arc length PQ = radius r,then angle POQ, = 1 radian)

(r)

(r) (Q)

(O)

Note:

Regardless of the size of the circle, radian as long the length of the arc is equal to the radius of the circle.

(s)

(r)In general, if the length of the arc is s units and the

radius is r units, then . That is, the size of the

angle is given by the ratio of the arc length to the

radius.

1.2 Relation between Radian and Degree measure

(sr)Consider the angle in a semicircle of radius r as shown below.

Since so we get

.

Then, we get radians.

In a semicircle, hence we get

1.2.1 Changing Degrees to Radians

Since , so .

e.g 1: Convert to radians

(Leave your answer in terms of unless asked for more accuracy)

e.g 2: Convert to radians, give your answer to 3 s.f.

e.g 3: Convert the following angles to degrees.

1.2.2 Changing Radians to Degrees

e.g 4: Convert radians to degrees.

e.g 5: Convert 2.1 radians to degrees.

e.g 6: Convert the following angles from radians to degrees.

Homework:

Ex 18A Pg 267 Qn 1 and Qn 2

2. Arc Length and Area of Sector2.1 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.

(l) (2l)

The arc length is directly proportional to the angle subtended at the center of the circle. We therefore have:

So Length of arc,

where is in radians.

(s) (r)

e.g 7: Find the length of the arc ABC.

cm or 10. 5cm

e.g 8: Find the radius of the sector ABC.

e.g 9: In a circle of radius 8 cm, find

a) the length of the arc which subtends an angle of radians at the centre,

b) the angle subtended by arc of length 6cm.

2.2 Area of a sector of a circle

In the diagram, the angle of sector is radians.

In degree:

In radian:

(s)

(A)

(r)

Now, as , we have:

The area, A, of a sector is given by:

where is in radians.

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 10: Find the area of sector ABC, where and r = 2cm, give your answer in terms of .

e.g 11: Find the area of sector ABC, where and r = 8cm, give your answer to 3 s.f.

Homework:

Ex 18A Pg 268 Qn 4

3. 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.

3.1 Area of a Triangle

The most common formula for the area of a triangle would be:

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 :

This tells us that the height, h, can be expressed as .

If we substitute this expression for the height in the original area of triangle formula, we can write:

Alternate version are:

or

e.g 12: In the figure, O is the centre of a circle of radius 6 cm and . Find

(a) the area of sector ,

(b) the area of .

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 . Find

(a) the area of the shaded region,

(b) the perimeter of the shaded region.

e.g 14: In the figure, O is the centre of the circle containing the sector . and are arcs of two concentric circles. .

If and , show that

(a) the area of the shaded region =

(b) the perimeter of the shaded region equals to

(c) Given that the shaded region is three quarters of the area of the sector , calculate the value of r.

(d) If, however, the total perimeter of the shaded region equals the total perimeter of the sector , find the value of .

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)

(b) the length of arc ,

(c) the area of the shaded region.

Homework:

Ex 18A Pg 268 Qn 5, 6, 7

4. Brief recap of Secondary level trigonometry4.1 Trigonometric Ratios of Acute Angles

The following formulae link the sides and angles in right-angled triangles:

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.

These formulae are often remembered using the acronym SOHCAHTOA or by using this : Silly Old Harry Couldnt Answer His Test On Algebra.

4.2 Sine Rule

The sine rule connects the length of sides and angles in any triangle .

It states that:

An alternate version of the formula is:

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.

4.3 Cosine Rule

The cosine rule also connects the length of sides and angles in any triangle

It states that

Equivalently, we also have these formulae:

Important: We can use Cosine rule when we are given

Two sides and the included angle.

Three sides.

e.g 16

5. Degree and Radian mode in calculator

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.

For instance, if you want to calculate , you must put your calculate in degree mode. Similarly, if you want the answer for , make sure your calculator is in radian mode before solving for the answer.

Pressing mode or shiftmode, should able to guide you to the mode setting screen in your calculator.

6. Trigonometric Ratios of Some Special Angles

Find the exact values of Trigonometric ratios. What this means is dont 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 and can be obtained using right angles triangles formed by taking half of a square and an equilateral triangle.

Angles,

Trigonometric types

5

H

A

O

x

A

B

C

c

a

b

p

p