Advertisements
Advertisements
Question
Solve the following:
`sin^-1x+sin^-1 2x=pi/3`
Solution
We know
`sin^-1x+sin^-1y=sin^-1[xsqrt(1-y^2)+ysqrt(1-x^2)]`
∴ `sin^-1x+sin^-1 2x=pi/3`
⇒ `sin^-1x+sin^-1 2x=sin^-1(sqrt3/2)`
⇒ `sin^-1x-sin^-1(sqrt3/2)=-sin^-1 2x`
⇒ `sin^-1[xsqrt(1-3/4)+sqrt3/2sqrt(1-x^2)]=-sin^-1 2x`
⇒ `sin^-1[x/2+sqrt3/2sqrt(1-x^2)]=sin^-1(-2x)`
⇒ `x/2+sqrt3/2sqrt(1-x^2)=-2x`
⇒ `x+sqrt3sqrt(1-x^2)=-4x`
⇒ `5x=-sqrt3sqrt(1-x^2)`
Squaring both the sides,
`25x^2=3-3x^2`
⇒ `28x^2=3`
⇒ `x=+-1/2sqrt(3/7)`
APPEARS IN
RELATED QUESTIONS
Find the principal values of the following:
`cos^-1(-1/sqrt2)`
Evaluate the following:
`cos^-1(cos4)`
Evaluate the following:
`sec^-1(sec pi/3)`
Evaluate the following:
`sec^-1(sec (2pi)/3)`
Evaluate the following:
`sec^-1(sec (5pi)/4)`
Evaluate the following:
`cosec^-1{cosec (-(9pi)/4)}`
Evaluate the following:
`cot^-1(cot pi/3)`
Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`
Evaluate:
`cot(sin^-1 3/4+sec^-1 4/3)`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x < 0
Evaluate: `cos(sin^-1 3/5+sin^-1 5/13)`
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
`tan^-1 2/3=1/2tan^-1 12/5`
Find the value of the following:
`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1
If x > 1, then write the value of sin−1 `((2x)/(1+x^2))` in terms of tan−1 x.
Write the value of tan−1x + tan−1 `(1/x)`for x > 0.
Write the value of tan−1 x + tan−1 `(1/x)` for x < 0.
Write the range of tan−1 x.
Write the value of cos−1 \[\left( \tan\frac{3\pi}{4} \right)\]
Write the value of cos−1 (cos 6).
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
If \[\tan^{- 1} (\sqrt{3}) + \cot^{- 1} x = \frac{\pi}{2},\] find x.
Write the value of \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right)\]
Write the principal value of \[\cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right)\]
Write the value of \[\tan\left( 2 \tan^{- 1} \frac{1}{5} \right)\]
Write the value of \[\tan^{- 1} \left\{ 2\sin\left( 2 \cos^{- 1} \frac{\sqrt{3}}{2} \right) \right\}\]
Write the value of \[\cos^{- 1} \left( \cos\frac{14\pi}{3} \right)\]
Write the value of `cot^-1(-x)` for all `x in R` in terms of `cot^-1(x)`
If \[\cos\left( \sin^{- 1} \frac{2}{5} + \cos^{- 1} x \right) = 0\], find the value of x.
Find the value of \[\tan^{- 1} \left( \tan\frac{9\pi}{8} \right)\]
The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is
2 tan−1 {cosec (tan−1 x) − tan (cot−1 x)} is equal to
sin\[\left[ \cot^{- 1} \left\{ \tan\left( \cos^{- 1} x \right) \right\} \right]\] is equal to
\[\tan^{- 1} \frac{1}{11} + \tan^{- 1} \frac{2}{11}\] is equal to
\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\]
If 2 tan−1 (cos θ) = tan−1 (2 cosec θ), (θ ≠ 0), then find the value of θ.
If \[\tan^{- 1} \left( \frac{1}{1 + 1 . 2} \right) + \tan^{- 1} \left( \frac{1}{1 + 2 . 3} \right) + . . . + \tan^{- 1} \left( \frac{1}{1 + n . \left( n + 1 \right)} \right) = \tan^{- 1} \theta\] , then find the value of θ.
Find the real solutions of the equation
`tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2`
tanx is periodic with period ____________.
