find-a-simple-expression-for-y-sin-1-sin-x-x-R-and-x-in-rad-

Question Number 128546 by mr W last updated on 08/Jan/21

f i n d a s i m p l e e x p r e s s i o n f o r

y = \sin^{- 1} (\sin x)

x \in R a n d x i n r a d .

Commented by liberty last updated on 08/Jan/21

\sin x = \sin y; y = x + 2 k π

where - \frac{π}{2} < y < \frac{π}{2}

Commented by mr W last updated on 08/Jan/21

e x a m p l e :

x = 5

h o w d o y o u g e t y = f (5) w i t h y o u r

f o r m u l a ?

Commented by john_santu last updated on 08/Jan/21

do you meant 5 in degress ? or radiant - ?

Commented by Olaf last updated on 08/Jan/21

f (5) = {(- 1)}^{⌊ \frac{5}{π} + \frac{1}{2} ⌋} (5 - ⌊ \frac{5}{π} + \frac{1}{2} ⌋ π)

⌊ \frac{5}{π} + \frac{1}{2} ⌋ \approx ⌊ 2, 091 ⌋ = 2

\Rightarrow f (5) = {(- 1)}^{2} (5 - 2 π) = 5 - 2 π

Answered by Olaf last updated on 08/Jan/21

\forall x \in R, \exists k \in Z ∖ x - k π \in [- \frac{π}{2}, + \frac{π}{2} [

\Leftrightarrow k π ⩽ x + \frac{π}{2} < (k + 1) π

\Leftrightarrow k ⩽ \frac{x}{π} + \frac{1}{2} < k + 1

k = ⌊ \frac{x}{π} + \frac{1}{2} ⌋ (integer part)

and \forall θ \in R, \forall q \in Z, \sin (θ + q π) = {(- 1)}^{q} \sin θ

\Rightarrow \sin x = \sin (x - k π + k π) = {(- 1)}^{k} \sin (x - k π)

\Rightarrow \arcsin (\sin x) = \arcsin ({(- 1)}^{k} \sin (x - k π))

= {(- 1)}^{k} \arcsin (\sin (x - k π))

and x - k π \in [- \frac{π}{2}, + \frac{π}{2} [

\Rightarrow \arcsin (\sin x) = {(- 1)}^{k} (x - k π)

Formula for \arcsin (\sin x) :

\arcsin (\sin x) = {(- 1)}^{⌊ \frac{x}{π} + \frac{1}{2} ⌋} (x - ⌊ \frac{x}{π} + \frac{1}{2} ⌋ π)

Commented by mr W last updated on 08/Jan/21

t h a n k s a l o t!

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