Advertisements
Advertisements
प्रश्न
Solve the equation for x:sin−1x+sin−1(1−x)=cos−1x
Advertisements
उत्तर
We have,
sin−1x+sin−1(1−x)=cos−1x
`sin^−1 x -cos^−1 x=-sin^−1 (1−x)`
`sin^−1 x -cos^−1 x=sin^−1 (x-1) ......................(1) [because sin^(-1)(-x)=-sin^-1x]`
`Put sin^-1 x=theta and cos^-1 x= phi`
`sin theta=x and cos phi=x`
`then cos theta=sqrt(1-sin^2theta) and sin phi=sqrt(1-cos^2 phi)`
`cos theta=sqrt(1-x^2) and sin phi =sqrt(1-x^2)`
Applying the formula:
`sin(theta-phi)=sin theta cos phi-cos theta sin phi` , we get
`sin(theta-phi)=x.x-sqrt(1-x^2)sqrt(1-x^2)`
`sin(theta-phi)=x^2-(1-x^2)`
`sin(theta-phi)=x^2-1+x^2`
`sin(theta-phi)=2x^2-1`
`(theta-phi)=sin^-1(2x^2-1)`
`sin^-1x - cos^-1 x=sin^-1(2x^2-1).............(2)`
From (1) and (2), we get
`sin^-1 (2x^2-1)= sin^-1 (x-1)`
`2x^2-x=0`
`x(2x-1)=0`
`x=0 or 2x-1=0`
`x=0 or x=1/2`
APPEARS IN
संबंधित प्रश्न
Write the value of `tan(2tan^(-1)(1/5))`
Prove that :
`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`
Solve the following for x:
`sin^(-1)(1-x)-2sin^-1 x=pi/2`
Find the domain of definition of `f(x)=cos^-1(x^2-4)`
Find the domain of `f(x)=cos^-1x+cosx.`
Find the principal values of the following:
`cos^-1(sin (4pi)/3)`
Evaluate the following:
`cos^-1{cos(-pi/4)}`
Evaluate the following:
`cosec^-1(cosec (13pi)/6)`
Evaluate the following:
`cot^-1{cot ((21pi)/4)}`
Write the following in the simplest form:
`cot^-1 a/sqrt(x^2-a^2),| x | > a`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x < 0
Solve the following equation for x:
tan−1(x + 2) + tan−1(x − 2) = tan−1 `(8/79)`, x > 0
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
Prove that:
`2sin^-1 3/5=tan^-1 24/7`
Prove that
`sin{tan^-1 (1-x^2)/(2x)+cos^-1 (1-x^2)/(2x)}=1`
Find the value of the following:
`tan^-1{2cos(2sin^-1 1/2)}`
Solve the following equation for x:
`2tan^-1(sinx)=tan^-1(2sinx),x!=pi/2`
Solve the following equation for x:
`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`
Prove that `2tan^-1(sqrt((a-b)/(a+b))tan theta/2)=cos^-1((a costheta+b)/(a+b costheta))`
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 cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]
Write the value of sin−1 (sin 1550°).
Evaluate sin
\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
If \[\sin^{- 1} \left( \frac{2a}{1 - a^2} \right) + \cos^{- 1} \left( \frac{1 - a^2}{1 + a^2} \right) = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right),\text{ where }a, x \in \left( 0, 1 \right)\] , then, the value of x is
If x > 1, then \[2 \tan^{- 1} x + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] is equal to
The domain of \[\cos^{- 1} \left( x^2 - 4 \right)\] is
Prove that : \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) = \frac{\pi}{4} + \frac{1}{2} \cos^{- 1} x^2 ; 1 < x < 1\].
Solve for x : {xcos(cot-1 x) + sin(cot-1 x)}2 = `51/50`
