untitled



3

1

3

tan 30

a

a

OM

$ PM

.

2

2

cosec 30

a

a

PM

$ OP

,

3

2

3

2

sec 30

a

a

OM

$ OP

3

3

cot 30

a

a

PM

$ OM

.

Similarly,

2

3

2

3

sin 60

a

a

OP

$ OM

,

2

1

2

cos 60

a

a

OP

$ PM

, 3

3

tan 60

a

a

PM

$ OM

3

2

3

2

cosec 60

a

a

OM

$ OP

,

2

2

sec 60

a

a

PM

$ OP

,

3

1

3

cot 60

a

a

OM

$ PM

.

Trigonometric ratio of the angle 45 $
Let, ‘XOZ 45 $ and P is a point on OZ.
DrawPMAOX. In right angled triangle
'OPM,‘POM 45 $
So,‘OPM 45 $
Therefore, PM OM=a (suppose)

Now, OP^2 OM^2 PM^2 =
2
a +
2
a =2
2
a

or,OP 2 a
From the definition of trigonometric ratios, we get

2

1

2

sin 45

a

a

OP

$ PM

,

2

1

2

cos 45

a

a

OP

$ OM

,tan 45 1

a

a

OM

$ PM

2

sin 45

1

cosec 45

q

$ , 2

cos 45

1

sec 45

q

$ , 1

tan 45

1

cot 45

q

$

9 ⋅7 Trigonometric ratios of complementary angles
We know, if the sum of two acute angles is 90 $, one of
them is called complementary angle to the other. For
example, 30 $and 60 $; 15 $and 75 $are complementary
angles to each other.
In general, the angles θ and ( 90 $θ) are
complementary angles to each other.