If x- y- z- - -1- 1- such that sin-1 x - sin-1 y - sin-1 z - 3 -2 - find the value of x2006 - y2007 - z2008 - 9x2006 - y2007 - z2008-

Given sin^ 1x+sin^ 1y+sin^ 1z=3π2 This is satisfied only when x=y=z =1 sin^ 1x has range [ π2, π2] Hence x^2006+x^2007+x^2008 9x^2006+x^ ...

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Byju's Answer

Standard XII

Mathematics

Integration of Trigonometric Functions

If x, y, z,...

Question

If $x, y, z, \in [- 1, 1]$ , such that ${sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}$ , find the value of $x^{2006} + y^{2007} + z^{2008} - \frac{9}{x^{2006} + y^{2007} + z^{2008}}$ .

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Solution

Given
${sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}$
This is satisfied only when $x = y = z = 1$
${sin}^{- 1} x$ has range $[\frac{- π}{2}, \frac{π}{2}]$
Hence
$x^{2006} + x^{2007} + x^{2008} - \frac{9}{x^{2006} + x^{2007} + x^{2008}}$
$3 - \frac{9}{3} = 0$ .

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Similar questions

$s i n^{- 1} x + s i n^{- 1} y + s i n^{- 1} z = \frac{3 π}{2},$ then the value of $x^{100} + y^{100} + z^{100} - \frac{9}{x^{101} + y^{101} + z^{101}} =$

Q. Assertion :If

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

,then

\frac{3 \sum_{r = 1}^{2008} (x^{r} + y^{r})}{2 \sum_{r = 1}^{2008} (x^{r} y^{r})} = 3

Reason:

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

is possible only if

x = y = z = 1

Q. If

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

, then find the minimum value of

x^{2} + y^{2} + z^{2}

Q. If

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

, then the value of

x^{100} + y^{100} + z^{100} - \frac{9}{x^{101} + y^{101} + z^{101}}

Q. If

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

, then

(x^{500} + y^{500} + z^{500}) - (x^{501} + y^{501} + z^{501})

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If x,y,z,∈[−1,1], such that sin−1x+sin−1y+sin−1z=3π2 , find the value of x2006+y2007+z2008−9x2006+y2007+z2008.

Givensin−1x+sin−1y+sin−1z=3π2This is satisfied only when x=y=z=1sin−1x has range [−π2,π2]Hencex2006+x2007+x2008−9x2006+x2007+x20083−93=0.

If $x, y, z, \in [- 1, 1]$ , such that ${sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}$ , find the value of $x^{2006} + y^{2007} + z^{2008} - \frac{9}{x^{2006} + y^{2007} + z^{2008}}$ .

Given
${sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}$
This is satisfied only when $x = y = z = 1$
${sin}^{- 1} x$ has range $[\frac{- π}{2}, \frac{π}{2}]$
Hence
$x^{2006} + x^{2007} + x^{2008} - \frac{9}{x^{2006} + x^{2007} + x^{2008}}$
$3 - \frac{9}{3} = 0$ .