Advertisements
Advertisements
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`
Advertisements
उत्तर
`cos^-1( x/a) +cos^-1 (y/b)=alpha`
`cos^-1(x/a) =alpha-cos^-1 (y/b)`
`=>cos {cos^-1 (x/a)}=cos{alpha-cos^-1 (y/b)}`
`=>x/a=cos alpha cos{cos^-1 (y/b)}+sinalpha sin{cos^-1(y/b)}`
`=>x/a=y/b cos alpha+sin alpha sqrt(1-(y/b)^2)`
`=>x/a - y/b cos alpha =sin alpha sqrt(1-(y/b)^2)`
Squaring both sides, we get
`(x/a - y/b cos alpha)^2={sin alpha sqrt(1-(y/b)^2)}^2`
`(x/a)^2+(y/b)^2cos^2 alpha- (2xy)/(ab) cos alpha=sin^2 alpha- sin^2 alpha(y/b)^2`
`(x/a)^2+(y/b)^2cos^2 alpha+sin^2 alpha(y/b)^2-(2xy)/(ab) cos alpha=sin^2 alpha`
`(x/a)^2+(y/b)^2(cos^2 alpha+sin^2 alpha)-(2xy)/(ab) cos alpha=sin^2 alpha`
`=>(x/a)^2-(2xy)/(ab) cos alpha+(y/b)^2=sin^2 alpha`
APPEARS IN
संबंधित प्रश्न
Find the value of the following: `tan(1/2)[sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))],|x| <1,y>0 and xy <1`
Show that:
`2 sin^-1 (3/5)-tan^-1 (17/31)=pi/4`
If sin [cot−1 (x+1)] = cos(tan−1x), then find x.
Find the domain of `f(x) =2cos^-1 2x+sin^-1x.`
Evaluate the following:
`cos^-1(cos3)`
Evaluate the following:
`tan^-1(tan12)`
Evaluate the following:
`sec^-1(sec (5pi)/4)`
Evaluate the following:
`sec^-1(sec (13pi)/4)`
Write the following in the simplest form:
`tan^-1{x+sqrt(1+x^2)},x in R `
Write the following in the simplest form:
`tan^-1{(sqrt(1+x^2)-1)/x},x !=0`
Evaluate the following:
`sin(sin^-1 7/25)`
Evaluate the following:
`cosec(cos^-1 3/5)`
Solve: `cos(sin^-1x)=1/6`
Evaluate:
`cos{sin^-1(-7/25)}`
Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`
Prove the following result:
`sin^-1 12/13+cos^-1 4/5+tan^-1 63/16=pi`
Solve the following equation for x:
`tan^-1 x/2+tan^-1 x/3=pi/4, 0<x<sqrt6`
Solve the following equation for x:
`tan^-1(2+x)+tan^-1(2-x)=tan^-1 2/3, where x< -sqrt3 or, x>sqrt3`
Prove that
`tan^-1((1-x^2)/(2x))+cot^-1((1-x^2)/(2x))=pi/2`
Show that `2tan^-1x+sin^-1 (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.
Solve the following equation for x:
`cos^-1((x^2-1)/(x^2+1))+1/2tan^-1((2x)/(1-x^2))=(2x)/3`
Write the value of sin−1
\[\left( \sin( -{600}°) \right)\].
If x < 0, y < 0 such that xy = 1, then write the value of tan−1 x + tan−1 y.
Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]
If sin−1 x − cos−1 x = `pi/6` , then x =
The value of \[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right)\] is
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find \[\frac{dy}{dx}\] When \[\theta = \frac{\pi}{3}\] .
Find the domain of `sec^(-1) x-tan^(-1)x`
The equation sin-1 x – cos-1 x = cos-1 `(sqrt3/2)` has ____________.
