Advertisements
Advertisements
Question
Prove `(9pi)/8 - 9/4 sin^(-1) 1/3 = 9/4 sin^(-1) (2sqrt2)/3`
Solution
L.H.S = `(9pi)/8 - 9/4 sin^(-1) 1/3`
`= 9/4 (pi/2 - sin^(-1) 1/3)`
`= 9/4 (cos^(-1) 1/3)` ....(1) `[sin^(-1)x + cos^(-1) x = pi/2]`
Now, let `cos^(-1) 1/3 = x` Then, `cos x = 1/3 => sin x = sqrt(1 - (1/3)^2) = (2sqrt2)/3`
`:. x = sin^(-1) (2sqrt2)/3 => cos^(-1) 1/3 = sin^(-1) (2sqrt2)/3`
:. L.H.S = `9/4 sin^(-1) (2(sqrt2))/3` = R.H.S
APPEARS IN
RELATED QUESTIONS
Prove that `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3`
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
Prove `tan^(-1) 2/11 + tan^(-1) 7/24 = tan^(-1) 1/2`
Write the following function in the simplest form:
`tan^(-1) (sqrt((1-cos x)/(1 + cos x))), x < pi`
Write the following function in the simplest form:
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt3 <= x a/sqrt3`
Prove that:
`cos^(-1) 4/5 + cos^(-1) 12/13 = cos^(-1) 33/65`
Solve the following equation:
`2 tan^(-1) (cos x) = tan^(-1) (2 cosec x)`
Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`
Find: ∫ sin x · log cos x dx
Find the value, if it exists. If not, give the reason for non-existence
`sin^-1 [sin 5]`
Prove that `sin^-1 3/5 - cos^-1 12/13 = sin^-1 16/65`
Prove that `tan^-1x + tan^-1y + tan^-1z = tan^-1[(x + y + z - xyz)/(1 - xy - yz - zx)]`
If tan–1x + tan–1y + tan–1z = π, show that x + y + z = xyz
Solve: `sin^-1 5/x + sin^-1 12/x = pi/2`
Evaluate `tan^-1(sin((-pi)/2))`.
Evaluate: `sin^-1 [cos(sin^-1 sqrt(3)/2)]`
Prove that cot–17 + cot–18 + cot–118 = cot–13
If `tan^-1x = pi/10` for some x ∈ R, then the value of cot–1x is ______.
If α ≤ 2 sin–1x + cos–1x ≤ β, then ______.
Evaluate `cos[cos^-1 ((-sqrt(3))/2) + pi/6]`
If 3 tan–1x + cot–1x = π, then x equals ______.
If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.
The value of the expression tan `(1/2 "cos"^-1 2/sqrt3)`
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
If tan-1 2x + tan-1 3x = `pi/4,` then x is ____________.
Simplest form of `tan^-1 ((sqrt(1 + cos "x") + sqrt(1 - cos "x"))/(sqrt(1 + cos "x") - sqrt(1 - cos "x")))`, `π < "x" < (3π)/2` is:
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
`"sin"^-1 (1 - "x") - 2 "sin"^-1 "x" = pi/2`
If `"sin" {"sin"^-1 (1/2) + "cos"^-1 "x"} = 1`, then the value of x is ____________.
The Simplest form of `cot^-1 (1/sqrt(x^2 - 1))`, |x| > 1 is
The set of all values of k for which (tan–1 x)3 + (cot–1 x)3 = kπ3, x ∈ R, is the internal ______.
If `tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36)` = then prove that 24x2 – 23x – 12 = 0
Solve for x: `sin^-1(x/2) + cos^-1x = π/6`
