Advertisements
Advertisements
प्रश्न
If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,` find the value of x2 + y2 + z2
Advertisements
उत्तर
We know that the maximum value of `sin^-1x. sin^-1y, sin^-1z is pi/2` and minimum value of `sin^-1x, sin^-1y, sin^-1z is pi/2`
Now,
For maximum value
LHS `=(sin^-1x)^2+(sin^-1y)^2+(sin^-1z)^2`
`=(pi/2)^2+(pi/2)^2+(pi/2)^2`
`=3/4pi^2=`RHS
and For minimum value
LHS `=(sin^-1x)^2+(sin^-1y)^2+(sin^-1z)^2`
`=(-pi/2)^2+(-pi/2)^2+(-pi/2)^2`
`=3/4pi^2` = RHS
Now, For maximum value
`sin^-1x=pi/2,sin^-1y=pi/2,sin^-1z=pi/2`
⇒ `x = sin pi/2,y=sin pi/2, z = sin pi/2`
⇒ x = 1, y = 1, z = 1
∴ x2 + y2 + z2 = 1 + 1 + 1 = 3
and for minimum value
`sin^-1x=-pi/2,sin^-1y=-pi/2,sin^-1z=-pi/2`
⇒ `x=sin(-pi/2),y=sin(-pi/2),z=sin(-pi/2)`
⇒ x = -1, y = -1, z = -1
∴ x2 + y2 + z2 = 1 + 1 + 1 = 3
APPEARS IN
संबंधित प्रश्न
Find the principal values of the following:
`cos^-1(-1/sqrt2)`
Find the principal values of the following:
`cos^-1(tan (3pi)/4)`
`sin^-1(sin (17pi)/8)`
Evaluate the following:
`cos^-1{cos (13pi)/6}`
Evaluate the following:
`cos^-1(cos3)`
Evaluate the following:
`sec^-1(sec (25pi)/6)`
Evaluate the following:
`cot^-1(cot (19pi)/6)`
Evaluate the following:
`cot^-1{cot ((21pi)/4)}`
Evaluate the following:
`sin(tan^-1 24/7)`
Evaluate the following:
`sin(sec^-1 17/8)`
Evaluate the following:
`cot(cos^-1 3/5)`
Evaluate:
`sec{cot^-1(-5/12)}`
`sin(sin^-1 1/5+cos^-1x)=1`
Prove the following result:
`tan^-1 1/4+tan^-1 2/9=sin^-1 1/sqrt5`
Solve the following equation for x:
`tan^-1 2x+tan^-1 3x = npi+(3pi)/4`
`2tan^-1 3/4-tan^-1 17/31=pi/4`
Solve the following equation for x:
`tan^-1 1/4+2tan^-1 1/5+tan^-1 1/6+tan^-1 1/x=pi/4`
Solve the following equation for x:
`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`
Prove that:
`tan^-1 (2ab)/(a^2-b^2)+tan^-1 (2xy)/(x^2-y^2)=tan^-1 (2alphabeta)/(alpha^2-beta^2),` where `alpha=ax-by and beta=ay+bx.`
If x < 0, then write the value of cos−1 `((1-x^2)/(1+x^2))` in terms of tan−1 x.
Write the value of tan−1\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
If x < 0, y < 0 such that xy = 1, then write the value of tan−1 x + tan−1 y.
Write the principal value of \[\tan^{- 1} 1 + \cos^{- 1} \left( - \frac{1}{2} \right)\]
Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]
Wnte the value of the expression \[\tan\left( \frac{\sin^{- 1} x + \cos^{- 1} x}{2} \right), \text { when } x = \frac{\sqrt{3}}{2}\]
If \[\cos\left( \tan^{- 1} x + \cot^{- 1} \sqrt{3} \right) = 0\] , find the value of x.
Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]
If \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}} \right)\] = α, then x2 =
2 tan−1 {cosec (tan−1 x) − tan (cot−1 x)} is equal to
The number of solutions of the equation \[\tan^{- 1} 2x + \tan^{- 1} 3x = \frac{\pi}{4}\] is
The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]
sin \[\left\{ 2 \cos^{- 1} \left( \frac{- 3}{5} \right) \right\}\] is equal to
If \[3\sin^{- 1} \left( \frac{2x}{1 + x^2} \right) - 4 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + 2 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) = \frac{\pi}{3}\] is equal to
In a ∆ ABC, if C is a right angle, then
\[\tan^{- 1} \left( \frac{a}{b + c} \right) + \tan^{- 1} \left( \frac{b}{c + a} \right) =\]
The value of \[\sin\left( 2\left( \tan^{- 1} 0 . 75 \right) \right)\] is equal to
Write the value of \[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\] .
Find the simplified form of `cos^-1 (3/5 cosx + 4/5 sin x)`, where x ∈ `[(-3pi)/4, pi/4]`
The value of sin `["cos"^-1 (7/25)]` is ____________.
