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17. θ

θ

θ θ

1 sin

1 sin

(tan sec )^2





 18. cotA.tanB.

cotB tanA

cotA tanB





19. secA tanA.
    1 sinA

1 sinA







20. cotA cosecA.
    secA 1

secA 1







21. If cosAsinA 2 cosA, prove that cosAsinA 2 sinA


22. If
    3

1

tanA , find the value of

cosecA secA

cosecA secA

2 2

2 2





.

23. If
    3

4

cosecAcotA , what is the value of cosecAcotA?

24. If
    a

b

cotA , find the value of

asinA bcosA

asinA bcosA





.

9 ⋅6 Trigonometric ratios of the angles 30 $, 45 $ and 60 $
We have learnt to draw the angles having the measurement of 30 $, 45 $and 60 $
geometrically. The actual values of the trigonometric ratios for all these angles can
be determined geometrically.
Trigonometric ratios of the angles 30 $ and 60 $
Let, ‘XOZ 30 $ and Pis a point on the side OZ.
DrawPMAOX and extend PM upto Q
such that MQ PM. Add O,Q and extend upto Z.

Now, between 'POM and 'QOM,PM QM,
OM is the common side and included ‘PMO
= included ‘QMO 90 $
? 'POM#'QOM

Therefore, ‘QOM ‘POM 30 $

and ‘OQM ‘OPM 60 $

Again, ‘POQ ‘POM‘QOM 30 $ 30 $ 60 $

? 'OPQ is an equilateral triangle.

If OP 2 a,PM PQ OP a
2

1

2

1

[since 'POQ is an equilateral triangle]

From right-angled 'OPM, we get,

OM OP^2 PM^2 4 a^2 a^2 3 a.
We find the trigonometric ratios :

?
2

1

2

sin 30

a

a

OP

$ PM

,

2

3

2

3

cos 30

a

a

OP

$ OM