If the maximum value of x which satisfies the inequality ^ - 1 (sin...

Let $t = sin x$
From the graphs of $\sin^{- 1} t$ and $\cos^{- 1} t$, we know that,

$\sin^{- 1} t \ge \cos^{- 1} t$ for $t \in \left[\dfrac{1}{\sqrt{2}}, 1\right]$

$⇒ sin x \in \left[\dfrac{1}{\sqrt{2}}, 1\right]$
Hence, $x \in \left(\dfrac{\pi}{2}, \dfrac{3\pi}{4}\right]$
Maximum value is $\dfrac{3\pi}{4} = \lambda$

So, $\dfrac{2\lambda}{3} = \dfrac{\pi}{2} = 1 . 57$