## Yahoo! 2 years ago Find x^2 + y^2+ z^2 - 4xyz if $\sin ^{-1}x+\sin ^{-1} y+\sin ^{-1}z = \frac{ 3\Pi }{ 2 }$?

1. shubhamsrg

hint : max value of sin^-1 x ?

2. Fazeelayaz

ye sawal kaha se liya hai

3. shubhamsrg

arre,,arent you getting the catch ? that was a great thing acc to me,,whats the max value of xin^-1 x? o.O

4. shubhamsrg

great hint* :P

5. Fazeelayaz

u mean max value of sinx-1 or sinx

6. shubhamsrg

sin^-1

7. shubhamsrg

inverse

8. Fazeelayaz

pi/2 + pi/2 +pi/2 do u mean that

9. shubhamsrg

yep..the only possible ans..

10. Fazeelayaz

hmm i don't think so

11. shubhamsrg

really,,what makes you argue ?

12. Fazeelayaz

try that formula sin-1x + sin-1y=sin-1(x(1-y^2)^1/2+y(1-x^2)^1/2)

13. Fazeelayaz

bring sins into one form like sin-1(.....)=3pi/2

14. Fazeelayaz

ok u are may be right

15. Fazeelayaz

i was just trying to confirm it

16. shubhamsrg

ofcorse am right,, suppose i ask you sinx +siny + sinz =3, then ofcorse its possible when all are =1

17. shubhamsrg

same with this case.. hence x=y=z =1

18. Yahoo!

@shubhamsrg Why Did u Take sin as Maximum ? Sorry My Connection was Out Of Order :)

19. Yahoo!

20. satellite73

@shubhamsrg provided you with the answer

21. mukushla

i go with @shubhamsrg$\sin ^{-1}x+\sin ^{-1} y+\sin ^{-1}z \le \frac{ 3\pi }{ 2 }$equality occurs when$\sin ^{-1}x=\sin ^{-1} y=\sin ^{-1}z = \frac{\pi }{ 2 }$

22. satellite73

you have $\sin^{-1}(x)+\sin^{-1}(y)+\sin^{-1}(z)=\frac{3\pi}{2}$ but the the very largest $$\sin^{-1}(x)$$ can be is $$\frac{\pi}{2}$$ so they must each be $$\frac{\pi}{2}$$, otherwise they cannot add to $$\frac{3\pi}{2}$$