Advertisements
Advertisements
प्रश्न
Prove the following:
`sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3π)/(4)`
उत्तर
Let `sin^-1(1/sqrt(2)) = α, "where" - pi/(2) ≤ α ≤ pi/(2)`
∴ sin α = `(1)/sqrt(2) = sin pi/(4)`
∴ α = `pi/(4) ...[∵ - pi/(2) ≤ pi/(4) ≤ pi/(2)]`
∴ `sin^-1(1/sqrt(2)) = pi/(4)` ...(1)
Let `sin^-1(sqrt(3)/2) = β, "where" - pi/(2) ≤ β ≤ pi/(2)`
∴ sin β = `sqrt(3)/(2) = sin pi/(3)`
∴ β = `pi/(3) ...[∵ - pi/(2) ≤ pi/(3) ≤ pi/(2)]`
∴ `sin^-1(sqrt(3)/2) = pi/(3)` ...(2)
L.H.S. = `sin^-1(1/sqrt(2)) - 3sin^-1(sqrt(3)/2)`
= `pi/(4) - 3(pi/3)` ...[By (1) and (2)]
= `pi/(4) - pi`
= `-(3pi)/(4)`
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Find the principal value of `cos^(-1) (sqrt3/2)`
Find the principal value of `tan^(-1) (-sqrt3)`
Find the principal value of `cos^(-1) (-1/2)`
Find the principal value of `cos^(-1) (-1/sqrt2)`
Find the value of the following:
`cos^(-1) (1/2) + 2 sin^(-1)(1/2)`
Find the value of the following:
`tan^(-1) (tan (7x)/6)`
Find the domain of the following function:
`f(x)sin^-1sqrt(x^2-1)`
Find the domain of `f(x)=cotx+cot^-1x`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of sin `(A/2)`.
In ΔABC, if a = 18, b = 24, c = 30 then find the values of cos `A/2`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of tan `A/2`
In ΔABC prove that `(b + c - a) tan "A"/(2) = (c + a - b)tan "B"/(2) = (a + b - c)tan "C"/(2)`.
In ΔABC prove that `sin "A"/(2). sin "B"/(2). sin "C"/(2) = ["A(ΔABC)"]^2/"abcs"`
Find the principal value of the following: `sin^-1 (1/2)`
Evaluate the following:
`"cosec"^-1(-sqrt(2)) + cot^-1(sqrt(3))`
Prove the following:
`sin^-1(-1/2) + cos^-1(-sqrt(3)/2) = cos^-1(-1/2)`
Prove the following:
`sin^-1(3/5) + cos^-1(12/13) = sin^-1(56/65)`
Prove the following:
`tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/4)`
sin−1x − cos−1x = `pi/6`, then x = ______
The principal value of cos−1`(-1/2)` is ______
If `sin(sin^-1(1/5) + cos^-1(x))` = 1, then x = ______
If tan−1x + tan−1y + tan−1z = π, then show that `1/(xy) + 1/(yz) + 1/(zx)` = 1
Prove that cot−1(7) + 2 cot−1(3) = `pi/4`
Show that `sin^-1(3/5) + sin^-1(8/17) = cos^-1(36/85)`
Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`
Show that `tan^-1 (1/2) + tan^-1 (2/11) = tan^-1 (3/4)`
Express `tan^-1 ((cos x - sin x)/(cos x + sin x))`, 0 < x < π in the simplest form.
Find the principal value of `tan^-1 (sqrt(3))`
A man standing directly opposite to one side of a road of width x meter views a circular shaped traffic green signal of diameter ‘a’ meter on the other side of the road. The bottom of the green signal Is ‘b’ meter height from the horizontal level of viewer’s eye. If ‘a’ denotes the angle subtended by the diameter of the green signal at the viewer’s eye, then prove that α = `tan^-1 (("a" + "b")/x) - tan^-1 ("b"/x)`
The value of cot `(tan^-1 2x + cot^-1 2x)` is ______
In ΔABC, tan`A/2 = 5/6` and tan`C/2 = 2/5`, then ______
The principle solutions of equation tan θ = -1 are ______
sin[3 sin-1 (0.4)] = ______.
If `sin^-1 3/5 + cos^-1 12/13 = sin^-1 P`, then P is equal to ______
If 2tan-1 (cos x) = tan-1 (cosec2 x), then x = ______.
`(sin^-1(-1/2) + tan^-1(-1/sqrt(3)))/(sec^-1 (-2/sqrt(3)) + cos^-1(1/sqrt(2))` = ______.
All trigonometric functions have inverse over their respective domains.
`"cos" 2 theta` is not equal to ____________.
When `"x" = "x"/2`, then tan x is ____________.
If sin-1 x – cos-1 x `= pi/6,` then x = ____________.
If tan-1 3 + tan-1 x = tan-1 8, then x = ____________.
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
If tan-1 (x – 1) + tan-1 x + tan-1 (x + 1) = tan-1 3x, then the values of x are ____________.
If 6sin-1 (x2 – 6x + 8.5) = `pi`, then x is equal to ____________.
`"sin" ["cot"^-1 {"cos" ("tan"^-1 "x")}] =` ____________.
Find the value of sec2 (tan-1 2) + cosec2 (cot-1 3) ____________.
`"tan"(pi/4 + 1/2 "cos"^-1 "x") + "tan" (pi/4 - 1/2 "cos"^-1 "x") =` ____________.
The equation 2cos-1 x + sin-1 x `= (11pi)/6` has ____________.
If `"x" in (- pi/2, pi/2), "then the value of tan"^-1 ("tan x"/4) + "tan"^-1 ((3 "sin" 2 "x")/(5 + 3 "cos" 2 "x"))` is ____________.
If tan-1 x – tan-1 y = tan-1 A, then A is equal to ____________.
`sin[π/3 - sin^-1 (-1/2)]` is equal to:
`"cos" ["tan"^-1 {"sin" ("cot"^-1 "x")}]` is equal to ____________.
If `"cot"^-1 (sqrt"cos" alpha) - "tan"^-1 (sqrt "cos" alpha) = "x",` then sinx is equal to ____________.
sin 6θ + sin 4θ + sin 2θ = 0, then θ =
The inverse of `f(x) = sqrt(3x^2 - 4x + 5)` is
If `(-1)/sqrt(2) ≤ x ≤ 1/sqrt(2)` then `sin^-1 (2xsqrt(1 - x^2))` is equal to
Domain and Rariges of cos–1 is:-
Find the value, if sin–1x = y, then `->`:-
`tan^-1 (1 - x)/(1 + x) = 1/2tan^-1x, (x > 0)`, x then will be equal to.
If f(x) = x5 + 2x – 3, then (f–1)1 (–3) = ______.
If f'(x) = x–1, then find f(x)
Number of values of x satisfying the system of equations `sin^-1sqrt(2 + e^(-2x) - 2e^-x) + sec^-1sqrt(1 - x^2 + x^4) = π/2` and `5^(1+tan^-1x)` = 4 + [cos–1x] is ______ (where [.] denotes greatest integer function)
Number of values of x which lie in [0, 2π] and satisfy the equation
`(cos x/4 - 2sinx) sinx + (1 + sin x/4 - 2cosx)cosx` = 0
`(tan^-1 (sqrt(3)) - sec^-1(-2))/("cosec"^-1(-sqrt(2)) + cos^-1(-1/2))` is equal to ______.
If y = `tan^-1 (sqrt(1 + x^2) - sqrt(1 - x^2))/(sqrt(1 + x^2) + sqrt(1 - x^2))`, then `dy/dx` is equal to ______.
Find the value of `cos(x/2)`, if tan x = `5/12` and x lies in third quadrant.
If sin–1x – cos–1x = `π/6`, then x = ______.
Find the value of `sin(2cos^-1 sqrt(5)/3)`.
If tan 4θ = `tan(2/θ)`, then the general value of θ is ______.
