Question

Find the exact value of sine [2 cosine Superscript negative 1 Baseline (- StartFraction StartRoot 2 EndRoot/2 EndFraction )] = nothing

Answer 1

Answer:

$\frac{3\pi}{2}$.

Step-by-step explanation:

The given expression is

$2\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

It can be rewritten as

$2\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)$

$2\left [\pi-\cos^{-1}\left(\frac{1}{\sqrt{2}}\right)\right]$            $[\because \cos^{-1}(-x)=\pi-\cos^{-1}(x), x\in [-1,1]]$

$2\left [\pi-\cos^{-1}\left(\cos \frac{\pi}{4}\right)\right]$

$2\left [\pi-\frac{\pi}{4}\right]$

$2\left [\frac{4\pi-\pi}{4}\right]$

$\frac{3\pi}{2}$

Hence, the exact value of given expression is $\frac{3\pi}{2}$.