vA mathematician knows how to solve a problem,

he can not solve it. – MILNE v

3.1 Introduction

The word ‘trigonometry’ is derived from the Greek words

‘trigon’ and ‘metron’ and it means ‘measuring the sides of

a triangle’. The subject was originally developed to solve

geometric problems involving triangles. It was studied by

sea captains for navigation, surveyor to map out the new

lands, by engineers and others. Currently, trigonometry is

used in many areas such as the science of seismology,

designing electric circuits, describing the state of an atom,

predicting the heights of tides in the ocean, analysing a

musical tone and in many other areas.

In earlier classes, we have studied the trigonometric

ratios of acute angles as the ratio of the sides of a right

angled triangle. We have also studied the trigonometric identities and application of

trigonometric ratios in solving the problems related to heights and distances. In this

Chapter, we will generalise the concept of trigonometric ratios to trigonometric functions

and study their properties.

3.2 Angles

Angle is a measure of rotation of a given ray about its initial point. The original ray is

Chapter

3

TRIGONOMETRIC FUNCTIONS

Arya Bhatt

(476-550)

Fig 3.1

Vertex

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50 MATHEMATICS

called the initial side and the final position of the ray after rotation is called the

terminal side of the angle. The point of rotation is called the vertex. If the direction of

rotation is anticlockwise, the angle is said to be positive and if the direction of rotation

is clockwise, then the angle is negative (Fig 3.1).

The measure of an angle is the amount of

rotation performed to get the terminal side from

the initial side. There are several units for

measuring angles. The definition of an angle

suggests a unit, viz. one complete revolution from the position of the initial side as

indicated in Fig 3.2.

This is often convenient for large angles. For example, we can say that a rapidly

spinning wheel is making an angle of say 15 revolution per second. We shall describe

two other units of measurement of an angle which are most commonly used, viz.

degree measure and radian measure.

3.2.1

Degree measure

If a rotation from the initial side to terminal side is

th

1

360

of

a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is

divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is

called a minute, written as 1′, and one sixtieth of a minute is called a second, written as 1″.

Thus, 1° = 60′, 1′ = 60″

Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are

shown in Fig 3.3.

Fig 3.2

Fig 3.3

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TRIGONOMETRIC FUNCTIONS 51

3.2.2 Radian measure

There is another unit for measurement of an angle, called

the radian measure. Angle subtended at the centre by an arc of length 1 unit in a

unit circle (circle of radius 1 unit) is said to have a measure of 1 radian. In the Fig

3.4(i) to (iv), OA is the initial side and OB is the terminal side. The figures show the

angles whose measures are 1 radian, –1 radian, 1

1

2

radian and –1

1

2

radian.

(i)

(ii)

(iii)

Fig 3.4 (i) to (iv)

(iv)

We know that the circumference of a circle of radius 1 unit is 2π. Thus, one

complete revolution of the initial side subtends an angle of 2π radian.

More generally, in a circle of radius r, an arc of length r will subtend an angle of

1 radian. It is well-known that equal arcs of a circle subtend equal angle at the centre.

Since in a circle of radius r, an arc of length r subtends an angle whose measure is 1

radian, an arc of length l will subtend an angle whose measure is

l

r

radian. Thus, if in

a circle of radius r, an arc of length l subtends an angle θ radian at the centre, we have

θ =

l

r

or l = r θ.

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