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

$α + β = 2 π$

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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