# Angle

- If rotation is anticlockwise, the angle is positive. If rotation is clockwise, the angle is negative. One full rotation indicates 360°.
- An angle is said to be acute angle if 0°
< 90°;

right angle if = 90°; obtuse angle if 90° < < 180°;

a straight angle if = 180°; and

a reflex angle if 180° < < 360°.

### There are three systems of measurement of an angle:

**Sexagesimal system**

In this system an angle is measured in degrees, minutes and seconds. A complete rotation describes 360°.

**1 right angle = 90°**(Since right angle is 1/4 th of full rotation)

A degree is further subdivided as

1degree = 60 minutes, written as 60'

and 1 minute = 60 seconds, written as 60''.

Thus 30·25° = 30° 15', 1·5' = 1' 30'' etc.

We say that 30·25° is in degrees notation; 30° 15' is in degree-minute-second notation.**Centesimal system**

In this system an angle is measured in grades, minutes and seconds.

Here**1 right angle = 100 grades**, written as 100g.

1 grade = 100 minutes, written as 100` and

1 minute = 100 seconds, written as 100``.

Thus 30·25g = 30g25`, 1·5` = 1`50`` etc.

### Circular system

In this system an angle is measured in radians. The circular measure of an angle is the number of radians it contains.*A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius. A radian is a constant angle.*

### Conversion Formula

radians = 2 right angles = 180° = 200 g.### Length of an arc of a circle

If an arc of length s subtends an angle radians at the center of a circle of radius r,then s = r

### Area of a sector of a circle

Area of sector = (1/2)r² = (1/2) s## Illustrative Examples

### Example

Express in degrees, radians as well as in grades the fourth angle of a quadrilateral, which has three angles 46° 30' 10'', 75° 44' 45'', 123° 9' 35'' respectively. (Take = 355/113)

### Solution

The sum of three given angles

= 46° 30' 10'' +75° 44' 45''+123° 9' 35''

= 245° 24' 30'' (since 90'' =1'30'' and 84' =1°24')

The sum of all four angles of quadrilateral = 360°.

Fourth angle = 360° -(245° 24' 30'') = 114° 35' 30''

(since 360° = 359° 59' 60'')

To convert it into radians,

114° 35' 30'' = 114° +(35 + 30/60)'

= 114°(71/2)' = 114° +(71/120)°

= (13751/120)°

= (13751/120) x (/180)
radians (As 180° = radians)

= (13751/120) x (1/180) x
(355/113) = 2 radians nearly

To convert the angle into centesimal system,

(13751/120)° = (13751/120) x (100/90) grades (since 90° = 100 grades)

= 127·3241 grades = 127g 32` 41``.

### Example

If G, D, denote respectively, the number of grades, degrees and radians in an angle, prove that

- G/100 = D/90 = 2 /
- G -D = 20 /

### Solution

We know that 1 right angle = 100g = 90° = /2 radians.

Let the angle be X right angles.

Then G = 100 X, D = 90 X, = (/2)
X ...(1)

- From (1), X = G/100 = D/90 = 2 /

Hence the result. - From (1), G -D = 100 X -90 X = 10 X

= 10 x 2 / = 20 /

## Exercise

- Draw diagrams for the following angles:

(i) -135° (ii) 740°. In which quadrant do they lie?

(iii) Find another positive angle whose initial and final positions are same as that of -135°, and indicate on the same diagram. - If lies in second quadrant, in which quadrant the
following will lie?

(i) /2 (ii) 2 (iii) - . - Express the following angles in radian measure as well as centesimal measure:

(i) 45° (ii) 40° 37' 30''. - The circular measure of an angle is 1·5. Express it in English as well as French system. Take = 3·14.
- The wheel of a carriage is 91 cms in diameter and makes 5 revolutions per second. How fast is the carriage running?
- A wheel makes 180 revolutions in a minute. Through how many radians does it turn in one second?
- Large hand of a clock is 21 cm long. How much distance does its extremity move in 20 minutes?
- Find the angle between the hands of a clock at 7.20 P.M.
- Sum of two angles is 80 grades and difference is 18°. Find the angles in degrees.
- The difference between two acute angles of a right-angled triangle is /3 in circular measure. Find these angles in degrees.
- The circular measures of two angles of a triangle are 1/2 and 1/3. Find the third angle in English system.
- The difference of two angles is 1° while their sum is 1 in circular measure. Find the angles in degrees in terms of .
- The angles of a triangle are in A.P. and the greatest is double the least. Find all the angles in circular system.
- Express the angle 236·345° in

(i>) degree-minute-second notation (ii) radians.

## Answers

**1.**(i) Third quadrant (ii) First quadrant (iii) 225°

**2.**(i) First quadrant (ii) Third or fourth quadrant

(iii) Third quadrant

**3.**(i) /4 radians; 50 grades

(ii) 65 /228 radians; 45 g 13` 89``

**4.**85° 59' 14'' or 95g 54` 14``

**5.**51· 48 km / hour

**6.**6

**7.**44 cm

**8.**100°

**9.**45° and 27°

**10.**63°, 27°

**11.**132° 16' 22''

**12.**(180 +)/2 , (180 +)/2

**13.**2/9, /3, 4/9 radians

**14.**(i) 236° 20' 42'' (ii) 4·125 radians