UPSEE Solved Model Paper 6

Here we can find which one is rational from given options.
We solve each options from using suitable trigonometrical identities and find which one is rational.
sin 15° = sin (45 - 30) = sin 45 cos 30 - cos 45 sin 30
=

1

√2

√3

2

−

1

√2

1

2

=

√3−1

2√2

Therefore, it is irrational.
(Since, Sin(A - B) = sinAcosB - cosAsinB)
Cos 150= Cos (45 – 30) = Cos 45 Cos 30 + Sin 45 Sin 30
=

1

√2

√3

2

+

1

√2

1

2

=

√3+1

2√2

Therefore, it is irrational
sin 15° cos 15°
=

1

2

(2sin15°cos15°)
=

1

2

sin30
=

1

2

×

1

2

=

1

4

Therefore, it is rational
sin 15° cos 75° = sin 15° cos (90 - 15)
= sin 5° . sin 15°
= sin215 = (

√3−1

2√2

)2
Therefore , it is irrational
sin 2A = 2 sin A cos A
cos (90 - 15) = sin 15°
Hence
sin 15° . cos 15° is a rational