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When the angle θ comes closer to 0 $, the length of the line segment PN reduces to

zero and in this case the value of
OP

PN

sinθ is approximately zero. At the same time,

the length of OP is equal to the length of ON and the value of cos T =

ON
OP is 1
approximately.
The angle, 0 $ is introduced for the convenience of discussion in trigonometry, and
the edge line and the original line of the angle 0 $ are supposed the same ray.
Therefore, in line with the prior discussion, it is said that, cos 0 $ 1 ,sin 0 $ 0.

Ifθis the acute angle, we see

θ

θ

θ

cos

sin

tan ,

θ

θ

θ

sin

cos

cot ,

θ

θ

cos

1

sec ,

θ

θ

sin

1

cosec ,

We define the angle 0 $in probable cases so that, those relations exists.

0

1

0

cos 0

sin 0

tan (^0) $
$
$
1
1
1
cos 0
1
sec 0 $ (^) $.
Since division by 0 is not allowed, cosec 0 $ and cot 0 $ can not be defined.
Again, when the angle θ is very closed to 90 $, hypotenuse OPis approximately
equal to PN. So the value of sinθ is approximately 1. On the other hand, if the
angle θ is is equal to 90 $,ON is nearly zero; the value of cosθ is approximately 0.
So, in agreement of formulae that are described above, we can say, cos 90 $ 0 ,
sin 90 $ 1.
0
1
0
sin 90
cos 90
cot (^90) $
$
$
1
1
1
sin 90
1
cosec 90 $ (^) $
Since one can not divided by 0, as before, tan 90 $ and sec 90 $ are not defined.
Observe : For convenience of using the values of trigonometric ratios of the angles
0 $, 30 $, 45 $, 60 $ and 90 $ are shown in the following table :