Advertisements
Advertisements
प्रश्न
Evaluate the following:
`tan 1/2(cos^-1 sqrt5/3)`
उत्तर
Let, `cos^-1 sqrt5/3=theta`
`=> costheta=sqrt5/3`
`=>2cos^2 theta/2-1=sqrt5/3`
`=>cos^2 theta/2=(3+sqrt5)/6`
`=>theta/2=cos^-1(sqrt((3+sqrt5)/6))`
`=tan^-1((sqrt(1-(sqrt((3+sqrt5)/6))^2))/(sqrt((3+sqrt5)/6)))`
`=tan^-1(sqrt(1-(3+sqrt5)/6)/sqrt(3+sqrt5/6))`
`=tan^-1((sqrt((3-sqrt5)/6))/(sqrt((3+sqrt5)/6)))`
`=tan^-1(sqrt((3-sqrt5)/(3+sqrt5)))`
`=tan^-1(sqrt(((3-sqrt5)(3-sqrt5))/((3+sqrt5)(3-sqrt5))))`
`=tan^-1(sqrt((3-sqrt5)^2/(9-5)))`
`=tan^-1((3-sqrt5)/2)`
i. e. , `1/2(cos^-1 sqrt5/3)=tan^-1 ((3-sqrt5)/2)`
`=>tan 1/2(cos^-1 sqrt5/3)=tan[tan^-1((3-sqrt5)/2)]`
`thereforetan 1/2(cos^-1 sqrt5/3)=(3-sqrt5)/2`
APPEARS IN
संबंधित प्रश्न
If `cos^-1( x/a) +cos^-1 (y/b)=alpha` , prove that `x^2/a^2-2(xy)/(ab) cos alpha +y^2/b^2=sin^2alpha`
Solve the following for x :
`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`
Solve the following for x:
`sin^(-1)(1-x)-2sin^-1 x=pi/2`
If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,` find the value of x2 + y2 + z2
`sin^-1(sin (17pi)/8)`
Evaluate the following:
`cos^-1{cos (13pi)/6}`
Evaluate the following:
`tan^-1(tan (6pi)/7)`
Evaluate the following:
`sec^-1(sec (25pi)/6)`
Evaluate the following:
`cosec^-1{cosec (-(9pi)/4)}`
Evaluate the following:
`cot^-1(cot pi/3)`
Evaluate the following:
`cot^-1{cot ((21pi)/4)}`
Write the following in the simplest form:
`tan^-1{x+sqrt(1+x^2)},x in R `
Evaluate the following:
`sin(sin^-1 7/25)`
Evaluate the following:
`sin(tan^-1 24/7)`
Prove the following result
`sin(cos^-1 3/5+sin^-1 5/13)=63/65`
Solve: `cos(sin^-1x)=1/6`
Evaluate:
`cosec{cot^-1(-12/5)}`
Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`
Evaluate:
`cos(sec^-1x+\text(cosec)^-1x)`,|x|≥1
Evaluate: `cos(sin^-1 3/5+sin^-1 5/13)`
Prove that: `cos^-1 4/5+cos^-1 12/13=cos^-1 33/65`
Find the value of the following:
`tan^-1{2cos(2sin^-1 1/2)}`
Solve the following equation for x:
`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`
For any a, b, x, y > 0, prove that:
`2/3tan^-1((3ab^2-a^3)/(b^3-3a^2b))+2/3tan^-1((3xy^2-x^3)/(y^3-3x^2y))=tan^-1 (2alphabeta)/(alpha^2-beta^2)`
`where alpha =-ax+by, beta=bx+ay`
Write the value of sin−1 (sin 1550°).
Write the value of cos \[\left( 2 \sin^{- 1} \frac{1}{2} \right)\]
If \[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2},\] then find x.
If x < 0, y < 0 such that xy = 1, then write the value of tan−1 x + tan−1 y.
Write the value of \[\tan^{- 1} \left\{ 2\sin\left( 2 \cos^{- 1} \frac{\sqrt{3}}{2} \right) \right\}\]
Write the principal value of \[\cos^{- 1} \left( \cos680^\circ \right)\]
The set of values of `\text(cosec)^-1(sqrt3/2)`
If \[\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = \alpha, then\frac{x^2}{a^2} - \frac{2xy}{ab}\cos \alpha + \frac{y^2}{b^2} = \]
sin \[\left\{ 2 \cos^{- 1} \left( \frac{- 3}{5} \right) \right\}\] is equal to
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find \[\frac{dy}{dx}\] When \[\theta = \frac{\pi}{3}\] .
Find : \[\int\frac{2 \cos x}{\left( 1 - \sin x \right) \left( 1 + \sin^2 x \right)}dx\] .
Find the domain of `sec^(-1) x-tan^(-1)x`
tanx is periodic with period ____________.
