Question
Question: If A = 2 tan<sup>–1</sup>(2\(\sqrt{2}\)– 1) and B = 3 sin<sup>–1</sup>\(\left( \frac{1}{3} \right)\)...
If A = 2 tan–1(22– 1) and B = 3 sin–1(31)+ sin–1(53), then-
A
A = B
B
A < B
C
A > B
D
None of these
Answer
A > B
Explanation
Solution
We have A = 2 tan–1(22– 1) = 2 tan–1 (1.828) ̃ A > 2 tan–1 3 ̃ A > 32π
Next sin– 1(31)< sin–1(21) ̃ sin– 1(31) < 6π
̃ sin– 131< 2π
Also 3 sin– 1 (31)= sin–1 [3.31−4(31)3]
= sin–1 (2723)= sin–1 (0. 852)
̃ 3 sin–1 (31)< sin–1 (23)̃ 3 sin–1(31) <3π
Further sin–1 (53)= sin–1 (0. 6) < sin–1 (23)
̃ sin–1(53)<3π
Hence, B = 3 sin–1(31)+ sin–1 (53)
<3π+ 3π=32π. Hence A > B