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CHAPTER 01
TRIGONOMETRY
Trigonometry is a branch of mathematics that deals with angles, triangles and the
trigonometric functions which have to do with the ratios between the sides of a right
angled triangle
...


hypotenuse
opposite

θ
adjacent

The subject matter of trigonometry is based on six trigonometric functions
...
These are
going to help us in calculating the sides and angles of a right angled triangle
...

(i)

cosec θ =

(ii)

sec θ

=

1
hypotenuse

sinθ
opposite
1
hypotenuse

cosθ
adjacent

8

cot θ

(iii)

=

1
adjacent

tanθ opposite

Let us consider a unit circle, i
...
a circle with radius 1 centered at the origin
...
An angle is positive if it is measured
in the counter-clockwise direction, and is negative if it is measured in the clockwise
direction
...

Then we can describe any angle by comparison with the circle
...
In a circle such a restriction is not there
...
The radian measure of an
angle does not refer to “degrees”, it refers to distance measured along an arc of the
unit circle
...

Consider the following diagram

1

1

1

The radian is a unit of angle measurement defined such that an angle of one radian
subtended from the center of a unit circle produces an arc with arc length one
...
Let
(x,y) denote the point at which this radial line intersects the unit circle x2 + y2 = 1
...

R

y

(x,y)
2

2

x +y =1
θ
0

10

(1,0)

x

Since the circumference of a circle is 2πr, where r is the radius of the circle, the
circumference of the unit circle(a circle with radius 1 unit) is 2π
...

From this follows that
360 
180 
1 radian =
=
≈ 57,3º
2

2

and
1º
=
≈ 0,0175 radians
...
If OP denotes a radial line,
then OP cuts an arc from Cr of length L
...
If θ = 360º, then L = 2πr
...
In fact

L
L

...

360 2π r
2π r
If θ is measured in radians, then

L

2 2π r
2π L L
or
θ =

...

There are two basic trigonometric functions, usually written as cosθ and sinθ
...

As θ varies, cosθ and sinθ oscillate between +1 and -1
...

Thus,

11

cos(θ+ 2π) = cosθ
and
sin(θ + 2π) = sinθ
In general, if n is the smallest positive integer such that f(x + n) = f(x), we say that the
function is periodic with period n
...

We also have a number of identities:
1
...


sin(- θ)

=

- sinθ

3
...


sin(π+θ)

=

- sinθ

5
...


sin(π -θ)

=

sinθ

7
...




sin    
2


=

cosθ

9
...




sin    
2


=

cosθ

11
...


sin2θ

= 2sinθcosθ

13
...


sin


2

=

1  cos 
2

15
...


sin2θ

=

1  cos 2
2

17
...


sinθ - sinβ

= 2 sin

19
...


cosθ + cosβ

= 2 cos

21
...


cos(θ + β)

=

cosθcos β - sinθ sinβ

23
...


tan(π + θ)

= tanθ

25
...


1 + cot2θ

= cosec2θ

27

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