Advertisements
Advertisements
प्रश्न
Show that `sin^-1(3/5) + sin^-1(8/17) = cos^-1(36/85)`
उत्तर
Let `sin^-1(3/5)` = x
∴ sin x = `3/5` and 0 < x < `pi/2`
∴ cos x > 0
Now, cos x = `sqrt(1 - sin^2x)`
= `sqrt(1 - (3/5)^2`
= `sqrt(1 - 9/25)`
= `4/5`
Let `sin^-1 (8/17)` = y
∴ sin y = `8/17` and 0 < y < `pi/2`
∴ cos y > 0
Now, cos y = `sqrt(1 - sin^2y)`
= `sqrt(1 - (8/17)^2`
= `sqrt(1 - 64/289)`
= `15/17`
But cos(x + y) = cosx cosy – sinx siny
= `4/5(15/17) - 3/15(8/17)`
= `(60 - 24)/85`
= `36/85`
∴ x + y = `cos^-1(36/85)`
∴ `sin^-1(3/5) + sin^-1(8/17) = cos^-1(36/85)`
APPEARS IN
संबंधित प्रश्न
Show that `2sin^-1(3/5) = tan^-1(24/7)`
Find the principal value of `cos^(-1) (-1/2)`
Find the value of the following:
`cos^(-1) (1/2) + 2 sin^(-1)(1/2)`
Prove that:
`tan^-1 ((sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))) = pi/4 - 1/2 cos^-1 x`, for `- 1/sqrt2 <= x <= 1`
[Hint: put x = cos 2θ]
Find the principal value of `sin^-1(1/sqrt2)`
Find the domain of the following function:
`f(x)=sin^-1x^2`
Find the domain of the following function:
`f(x)=sin^-1x+sin^-1 2x`
If `sin^-1 x + sin^-1 y+sin^-1 z+sin^-1 t=2pi` , then find the value of x2 + y2 + z2 + t2
Evaluate the following:
`tan^-1(-1/sqrt3)+tan^-1(-sqrt3)+tan^-1(sin(-pi/2))`
Find the set of values of `cosec^-1(sqrt3/2)`
Find the domain of `f(x)=cotx+cot^-1x`
Evaluate the following:
`tan^-1(-1/sqrt3)+cot^-1(1/sqrt3)+tan^-1(sin(-pi/2))`
Solve for x:
`tan^-1 [(x-1),(x-2)] + tan^-1 [(x+1),(x+2)] = x/4`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of A(ΔABC)
In ΔABC prove that `sin "A"/(2). sin "B"/(2). sin "C"/(2) = ["A(ΔABC)"]^2/"abcs"`
Find the principal value of the following: tan-1(– 1)
Find the principal value of the following: cos- 1`(-1/2)`
Evaluate the following:
`tan^-1 sqrt(3) - sec^-1 (-2)`
Prove the following:
`sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3π)/(4)`
Prove the following:
`sin^-1(3/5) + cos^-1(12/13) = sin^-1(56/65)`
Prove the following:
`2tan^-1(1/3) = tan^-1(3/4)`
Prove the following:
`tan^-1[sqrt((1 - cosθ)/(1 + cosθ))] = θ/(2)`, if θ ∈ (– π, π).
Find the principal solutions of the following equation:
tan 5θ = -1
Find the principal solutions of the following equation:
cot 2θ = 0.
The principal value of sin−1`(1/2)` is ______
`tan^-1(tan (7pi)/6)` = ______
Evaluate cot(tan−1(2x) + cot−1(2x))
Prove that `2 tan^-1 (3/4) = tan^-1(24/7)`
Prove that cot−1(7) + 2 cot−1(3) = `pi/4`
Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`
Prove that:
`tan^-1 (4/3) + tan^-1 (1/7) = pi/4`
Express `tan^-1 [(cos x)/(1 - sin x)], - pi/2 < x < (3pi)/2` in the simplest form.
Choose the correct alternative:
cos 2θ cos 2ϕ+ sin2 (θ – ϕ) – sin2 (θ + ϕ) is equal to
The value of cot `(tan^-1 2x + cot^-1 2x)` is ______
If `sin^-1(x/13) + cosec^-1(13/12) = pi/2`, then the value of x is ______
sin[3 sin-1 (0.4)] = ______.
If `sin^-1x + cos^-1y = (3pi)/10,` then `cos^-1x + sin^-1y =` ______
If `sin^-1 3/5 + cos^-1 12/13 = sin^-1 P`, then P is equal to ______
The principal value of `tan^{-1(sqrt3)}` is ______
If sin `(sin^-1 1/3 + cos^-1 x) = 1`, then the value of x is ______.
In a triangle ABC, ∠C = 90°, then the value of `tan^-1 ("a"/("b + c")) + tan^-1("b"/("c + a"))` is ______.
`cos(2sin^-1 3/4+cos^-1 3/4)=` ______.
The domain of the function y = sin–1 (– x2) is ______.
The domain of y = cos–1(x2 – 4) is ______.
The equation tan–1x – cot–1x = `(1/sqrt(3))` has ______.
Prove that `cot(pi/4 - 2cot^-1 3)` = 7
Show that `2tan^-1 (-3) = (-pi)/2 + tan^-1 ((-4)/3)`
If 2 tan–1(cos θ) = tan–1(2 cosec θ), then show that θ = π 4, where n is any integer.
`"cos" 2 theta` is not equal to ____________.
`("cos" 8° - "sin" 8°)/("cos" 8° + "sin" 8°)` is equal to ____________.
If `"sin"^-1("x"^2 - 7"x" + 12) = "n"pi, AA "n" in "I"`, then x = ____________.
If tan-1 3 + tan-1 x = tan-1 8, then x = ____________.
If tan-1 (x – 1) + tan-1 x + tan-1 (x + 1) = tan-1 3x, then the values of x are ____________.
`"cos"^-1 ["cos" (2 "cot"^-1 (sqrt2 - 1))] =` ____________.
If `"cot"^-1 (sqrt"cos" alpha) - "tan"^-1 (sqrt "cos" alpha) = "x",` then sinx is equal to ____________.
If A = `[(cosx, sinx),(-sinx, cosx)]`, then A1 A–1 is
`2tan^-1 (cos x) = tan^-1 (2"cosec" x)`, then 'x' will be equal to
Value of `sin(pi/3 - sin^1 (- 1/2))` is equal to
What is the values of `cos^-1 (cos (7pi)/6)`
Assertion (A): The domain of the function sec–12x is `(-∞, - 1/2] ∪ pi/2, ∞)`
Reason (R): sec–1(–2) = `- pi/4`
Let x = sin–1(sin8) + cos–1(cos11) + tan–1(tan7), and x = k(π – 2.4) for an integer k, then the value of k is ______.
cos–1(cos10) is equal to ______.
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
If x ∈ R – {0}, then `tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2)))`
The value of `cos^-1(cos(π/2)) + cos^-1(sin((2π)/2))` is ______.
If cos–1 x > sin–1 x, then ______.
Find the value of `cos(x/2)`, if tan x = `5/12` and x lies in third quadrant.
The value of `tan(cos^-1 4/5 + tan^-1 2/3)` is ______.
Find the value of `sin(2cos^-1 sqrt(5)/3)`.
Find the value of `tan^-1(x/y) + tan^-1((y - x)/(y + x))`
