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Property 2

Solve : sin...

Question

Solve : $s i n^{- 1} (\frac{s i n + c o s x}{\sqrt{2}}) \frac{- π}{4} < x < \frac{π}{4}$

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Solution

${sin}^{- 1} (\frac{s i n x + c o s x}{\sqrt{2}})$

$= {sin}^{- 1} (\frac{s i n x}{\sqrt{2}} + \frac{c o s x}{\sqrt{2}})$

$= {sin}^{- 1} (c o s \frac{π}{4} \cdot s i n x + s i n \frac{π}{4} \cdot c o s x)$

$= {sin}^{- 1} {s i n (x + \frac{π}{4})}$

$= x + \frac{π}{4}$

$- \frac{π}{4} < x < \frac{π}{4}$ (given)

So,
$0 < x + \frac{π}{4} < \frac{π}{2}$

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{sin}^{- 1} {\frac{sin x + cos x}{\sqrt{2}}}, - \frac{π}{4} < x < \frac{π}{4} .

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m

lim x \to \frac{π}{4} \frac{\sqrt{2} - cos x - sin x}{{(\frac{π}{4} - x)}^{2}}

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