TS EAMCET 19-July-2022 Shift 2 Solved Paper

Given, tan‌θ=−‌

3

4

If θ does not lie in second quadrant, then θ lies in fourth quadrant as tan‌θ is negative only in second and fourth quadrant.
θ∈(‌

3π

2

,2π)⇒‌

θ

2

∈(‌

3π

4

,π)
⇒‌‌2θ∈(3π,4π)
⇒θ∕2 lies in second quadrant.
2θ lies in third and fourth quadrant.
tan‌θ=−‌

3

4

⇒‌

2‌tan‌ θ 2

1−tan2‌ θ 2

=−‌

3

4

⇒8‌tan‌

θ

2

=−3+3tan2‌

θ

2

⇒3tan2‌

θ

2

−8‌tan‌

θ

2

−3=0
⇒3tan2‌

θ

2

−9‌tan‌

θ

2

+tan‌

θ

2

−3=0
⇒‌‌tan‌

θ

2

=−‌

1

3

‌‌[∵tan‌

θ

2

≠3]
sin‌2θ=‌

2‌tan‌θ

1+tan2θ

=‌

2(− 3 4 )

1+‌ 9 16

=−‌

3

2

×‌

16

25

=−‌

24

25