Question

Evaluate the following

${sin}^{ - 1} (sin5)$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given inverse Trigonometric function is

$\rm :\longmapsto\: {sin}^{ - 1}(sin5)$

We know,

$\boxed{\tt{ {sin}^{ - 1}(sinx) = x \: \: if \: x \in \: \bigg[ - \dfrac{\pi}{2},\dfrac{\pi}{2}\bigg]}}$

And

$\boxed{\tt{5 \: \: \cancel \in \: \bigg[ - \dfrac{\pi}{2},\dfrac{\pi}{2}\bigg]}}$

We know,

$\purple{\rm :\longmapsto\: {1}^{c} \approx57 \degree}$

So,

$\purple{\rm :\longmapsto\: {5}^{c} \approx285 \degree}$

$\rm\implies \:\dfrac{3\pi}{2} < 5 < 2\pi$

$\rm\implies \: - \dfrac{3\pi}{2} > - 5 > - 2\pi$

$\rm\implies \:2\pi - \dfrac{3\pi}{2} > 2\pi- 5 > 2\pi - 2\pi$

$\rm\implies \: \dfrac{\pi}{2} > 2\pi- 5 > 0$

$\bf\implies \:0 < 2\pi - 5 < \dfrac{\pi}{2}$

So,

$\bf\implies \:sin(2\pi - 5) = - sin5$

$\bf\implies \: sin5 = - sin(2\pi - 5)$

So, Given expression

$\rm :\longmapsto\: {sin}^{ - 1}(sin5)$

can be rewritten as

$\rm \:  =  \: \: {sin}^{ - 1}\bigg[ - sin(2\pi - 5) \bigg]$

$\rm \:  =  \: \: - {sin}^{ - 1}\bigg[sin(2\pi - 5) \bigg]$

$\rm \:  =  \: - (2\pi - 5)$

$\rm \:  =  \: 5 - 2\pi$

Hence,

$\purple{\rm\implies \:\boxed{\tt{ {sin}^{ - 1}(sin5) = 5 - 2\pi \: }}}$

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MORE TO KNOW

$\boxed{\tt{ {sin}^{ - 1}(sinx) = x \: \: if \: x \in \: \bigg[ - \dfrac{\pi}{2},\dfrac{\pi}{2}\bigg]}}$

$\boxed{\tt{ {tan}^{ - 1}(tanx) = x \: \: if \: x \in \: \bigg( - \dfrac{\pi}{2},\dfrac{\pi}{2}\bigg)}}$

$\boxed{\tt{ {cos}^{ - 1}(cosx) = x \: \: if \: x \in \: \bigg[0, \: \pi\bigg]}}$