If ∑$$i=12n$$ $$sin$$−$$1$$⁡$$xi=n$$π, then ∑$$i=12n$$ xi is equal to

Question

If ∑$$i=12n$$ $$sin$$−$$1$$⁡$$xi=n$$π, then ∑$$i=12n$$ xi is equal to

Infinity Learn · Accepted Answer

We have, ∑$$i=12n$$ $$sin$$−$$1$$⁡$$xi=n$$π⇒$$sin$$−$$1$$⁡$$x1+$$$$sin$$−$$1$$⁡$$x2+$$…$$+$$$$sin$$−$$1$$⁡$$x2n=n$$$$π$$Let $$sin$$−$$1$$⁡$$x1=$$α$$1$$,$$sin$$−$$1$$⁡$$x2=$$α$$2$$,…,$$sin$$−$$1$$⁡$$xn=$$αn∴α$$1+$$α$$2+$$…$$+$$α$$2n=n$$π                 ……… (1)⇒$$x1=$$$$sin$$⁡α$$1$$,$$x2=$$$$sin$$⁡α$$2$$,…,$$x2n=$$$$sin$$⁡α$$2n$$∴∑$$i=12n$$ $$xi=x1+x2+$$…$$+x2n=$$$$sin$$⁡α$$1+$$$$sin$$⁡α$$2+$$…$$+$$$$sin$$⁡α$$2nClearly$$, from (1), α$$1=$$π$$2$$,∀$$i=1$$,$$2$$,…,$$2n$$∴ ∑$$i=12n$$ $$xi=1+1+$$…$$+1$$⏟$$2n$$ times $$=2n$$

If $\sum_{i = 1}^{2 n} \sin^{- 1} x_{i} = nπ$ , then $\sum_{i = 1}^{2 n} x_{i}$ is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If ∑i=12n sin−1⁡xi=nπ, then ∑i=12n xi is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If $\sum_{i = 1}^{2 n} \sin^{- 1} x_{i} = nπ$ , then $\sum_{i = 1}^{2 n} x_{i}$ is equal to