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Trigonometric ratios of complementary angles
Let, ‘XOY θ and P is the point on the side OY of the angle. We
drawPMAOX.
Since the sum of the three angles of a triangle is two right angles therefore, in the
right angled triangle POM,‘PMO 90 $
and‘OPM‘POM one right angle = 90 $
? ‘OPM 90 $‘POM 90 $θ
[Since‘POM ‘XOY θ]

?
OP

OM

sin ( 90 $θ) =cos‘POM=cosθ

OP

PM

cos( 90 $θ) =sin‘POM =sinθ

PM

OM

tan( 90 $θ) =cot‘POM=cotθ

OM

PM

cot( 90 $θ) =tan‘POM=tanθ

PM

OP

sec( 90 $θ) =cosec‘POM=cosecθ

OM

OP

cosec( 90 $θ) =sec‘POM=secθ.

We can express the above formulae in words below :
sine of complementary angle =cosine of angle
cosine of complementary angle =sine of angle
tangent of complementary angle =cotangent of angle etc.

Activity : 1. If

3

5

sec( 90 $θ) ,find the value of cosecθcotθ.

9 ⋅8 Trigonometric ratios of the angles 0 $ and 90 $
We have learnt how to determinal the trigonometric ratios for
the acute angle θ of a right angled triangle. Now, we see, if the
angle is made gradually smaller, how the trigonometric ratios
change. As θ get smallers the length of the side PN also gets
smaller. The point P closes to the point N and finally the
angle θ comes closer to the angle 0 $,OP is reconciled with
ON approximately.