Angle

A measure formed between two rays having a common initial point is called an angle. The two rays are called the arms or sides of the angle and the common initial point is called the vertex of the angle.

In the above figure OA is said to be ‘initial side’ and the other ray is said to be ‘terminal side’.

SYSTEMS OF MEASUREMENT OF ANGLES:

Sexagesimal system: In this system a right angle is divided into 90 equal parts called degrees. Each degree is divided into 60 equal parts called minutes and each minute is divided into 60 equal parts called seconds.

Thus, 1 right angle = 90 degrees ( 90°)

1° = 60 minutes (60')

1’ = 60 seconds (60").

Centesimal system: In this system a right angle is divided into 100 equal parts, called grades. Each grade is sub divided into 100 minutes, and each minute into 100 seconds.

Thus, 1 right angle = 100 grades (100g)

1 grade = 100 minutes (100')

1 minute =100 seconds (100").

Circular system: In this system the unit of measurement is radian. One radian, written as 1o, is the measure of an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.

The number of radians in an angle subtended by an arc of a circle at the centre is equal lenght of arc/radius

∴ θ = s/r

Where, θ = angle in radian, s = arc length and r = radius.

RELATION BETWEEN THREE SYSTEM OF MEASUREMENT OF ANGLES:

D/90 = G/100 = 2R/π

Where, D = number of degrees,

G = number of grades,

and R = number of radians.

question 1. Express 110o 30’ in radians.

Solution: 110o30’ = 110°

question 2. Express in degrees: (a) (2π/15)°, (b) (–2)c

Solution: (a)

(b)

= –114o

= –114o 32’ = – 114o 32’ 44”.

question 3. One angle of a triangle is 2x/3 grades another is 3x/2 degrees, whilst the third is πx/375 radians; express them all in degrees.

Solution:

But

∴ 6xo + 15xo + 24xo = 1800^o

45xo = 1800^o

x = 40^o

Hence, three angles of the triangle are 24°, 60° and 96°.

question 4. The angles of a triangle are in A.P. and the number of degrees in the least is to the number of radians in the greatest as 60 to πc. Find the angles in degrees.

Solution: The three angles in A. P.; if y is common difference, let these angles be (x + y)°, x° and

( x - y)^o

∴ x + y + x + x – y = 180°

∴ x = 60°

According to the question,

or 3 (x - y) = x + y

or 4y = 2x

or y = x/2

∴ y = 60°/2 = 30°

Hence three angles are 30°, 60° and 90°.