First slide

Properties of ITF

Question

If $α = \sin^{- 1} \frac{\sqrt{3}}{2} + \sin^{- 1} \frac{1}{3}$ and $β = \cos^{- 1} \frac{\sqrt{3}}{2} + \cos^{- 1} \frac{1}{3}$ , then

Moderate

A

α > β
B

α = β
C

α < β
D

α + β = 2π

Solution

From given equations, it can be seen that
α + β = π
Since, $\sin^{- 1} x + \cos^{- 1} x = \frac{π}{2} \forall x$
Also, $α = \frac{π}{3} + \sin^{- 1} \frac{1}{3} < \frac{π}{3} + \sin^{- 1} \frac{1}{2}$
as sinθ is increasing in $[0, \frac{π}{2}]$
$∴ α < \frac{π}{3} + \frac{π}{6} = \frac{π}{2} \Rightarrow β > \frac{π}{2} > α \Rightarrow α < β$

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