Find the value of sin -1 sin3-5 -

Given : sin ^ 1 ( sin3π5 ) We know that sin ^ 1(sin x)=x,x∈ [ π2,π2 ] Since 3π5∉ [ π2,π2 ], nbsp;which is the range of principal value branch of sin ^ 1 How ...

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Standard XII

Mathematics

Sum of Cosines of Angles in Arithmetic Progression

Find the valu...

Question

Find the value of ${sin}^{- 1} (sin \frac{3 π}{5}) .$

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Solution

Given :
${sin}^{- 1} (sin \frac{3 π}{5})$
We know that ${sin}^{- 1} (sin x) = x, x \in [- \frac{π}{2}, \frac{π}{2}]$
Since $\frac{3 π}{5} \notin [- \frac{π}{2}, \frac{π}{2}]$ , which is the range of principal value branch of ${sin}^{- 1}$
However, $sin (\frac{3 π}{5}) = sin (π - \frac{3 π}{5}) = sin \frac{2 π}{5}$
and $\frac{2 π}{5} \in [- \frac{π}{2}, \frac{π}{2}]$
$∴ {sin}^{- 1} (sin \frac{3 π}{5}) = {sin}^{- 1} (sin \frac{2 π}{5}) = \frac{2 π}{5}$

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Sum of Cosines of Angles in Arithmetic Progression

Standard XII Mathematics

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Find the value of sin−1(sin3π5).

Given : sin−1(sin3π5) We know that sin−1(sinx)=x,x∈[−π2,π2] Since 3π5∉[−π2,π2], which is the range of principal value branch of sin−1 However, sin(3π5)=sin(π−3π5)=sin2π5 and 2π5∈[−π2,π2] ∴sin−1(sin3π5)=sin−1(sin2π5)=2π5

Find the value of ${sin}^{- 1} (sin \frac{3 π}{5}) .$