Advertisements
Advertisements
प्रश्न
Evaluate the following:
`cos^-1(1/2) + 2sin^-1(1/2)`
उत्तर
Let `cos^-1(1/2)` = α, where 0 ≤ α ≤ π
∴ cos α = `1/2 = cos (pi)/(3)`
∴ α = `pi/(3) ...[∵ 0 < pi/(3) < pi]`
∴ `cos^-1(1/2) = pi/(3)` ...(1)
Let `sin^-1(1/2) = β, "where" (-pi)/(2) ≤ β ≤ pi/(2)`
∴ sin β = `(1)/(2) = sin (pi)/(6)`
∴ β = `pi/(6) ...[∵ (-pi)/(2) ≤ pi/(6) ≤ pi/(2)]`
∴ `sin^-1(1/2) = pi/(6)` ...(2)
`cos^-1(1/2) = pi/(3) and sin^-1(1/2) = pi/(6)`
∴ `cos^-1(1/2) + 2sin^-1(1/2)`
= `pi/(3) + 2(pi/6)`
= `pi/(3) + pi/(3)`
= `(2pi)/(3)`.
APPEARS IN
संबंधित प्रश्न
Find the principal values of `sin^(-1) (-1/2)`
Find the principal value of `cos^(-1) (sqrt3/2)`
Find the principal value of cosec−1 (2)
Find the principal value of `tan^(-1) (-sqrt3)`
Find the principal value of `cot^(-1) (sqrt3)`
Find the principal value of `cos^(-1) (-1/sqrt2)`
Find the principal value of `cosec^(-1)(-sqrt2)`
Find the value of the following:
`tan^(-1) (tan (7x)/6)`
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]
Evaluate the following:
`tan^-1 1+cos^-1 (-1/2)+sin^-1(-1/2)`
Evaluate the following:
`tan^-1(tan (5pi)/6)+cos^-1{cos((13pi)/6)}`
Find the set of values of `cosec^-1(sqrt3/2)`
Evaluate the following:
`tan^-1(-1/sqrt3)+cot^-1(1/sqrt3)+tan^-1(sin(-pi/2))`
Prove that:
cot−1 7 + cot−1 8 + cot−1 18 = cot−1 3 .
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 cosA
Find the principal value of the following: `sin^-1 (1/2)`
Find the principal value of the following: cos- 1`(-1/2)`
Evaluate the following:
`tan^-1(1) + cos^-1(1/2) + sin^-1(1/2)`
Evaluate the following:
`tan^-1 sqrt(3) - sec^-1 (-2)`
Evaluate the following:
`"cosec"^-1(-sqrt(2)) + cot^-1(sqrt(3))`
Prove the following:
`sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3π)/(4)`
Prove the following:
`sin^-1(-1/2) + cos^-1(-sqrt(3)/2) = cos^-1(-1/2)`
Prove the following:
`tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/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:
sin 2θ = `− 1/(sqrt2)`
Find the principal solutions of the following equation:
tan 5θ = -1
Find the principal solutions of the following equation:
cot 2θ = 0.
`tan^-1(tan (7pi)/6)` = ______
Prove that `2 tan^-1 (3/4) = tan^-1(24/7)`
Prove that sin `[tan^-1 ((1 - x^2)/(2x)) + cos^-1 ((1 - x^2)/(1 + x^2))]` = 1
Show that `sin^-1(3/5) + sin^-1(8/17) = cos^-1(36/85)`
Find the principal value of the following:
cosec-1 (2)
Evaluate:
`cos[tan^-1 (3/4)]`
Evaluate: sin`[1/2 cos^-1 (4/5)]`
Prove that `tan^-1 (m/n) - tan^-1 ((m - n)/(m + n)) = pi/4`
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)`
In Δ ABC, with the usual notations, if sin B sin C = `"bc"/"a"^2`, then the triangle is ______.
Which of the following function has period 2?
`sin^2(sin^-1 1/2) + tan^2 (sec^-1 2) + cot^2(cosec^-1 4)` = ______.
In a triangle ABC, ∠C = 90°, then the value of `tan^-1 ("a"/("b + c")) + tan^-1("b"/("c + a"))` is ______.
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then θ = ______
`cos^-1 4/5 + tan^-1 3/5` = ______.
The value of `sin^-1[cos(pi/3)] + sin^-1[tan((5pi)/4)]` is ______.
When `"x" = "x"/2`, then tan x is ____________.
`"sin"^2 25° + "sin"^2 65°` is equal to ____________.
`("cos" 8° - "sin" 8°)/("cos" 8° + "sin" 8°)` is equal to ____________.
If sin-1 x – cos-1 x `= pi/6,` then x = ____________.
If tan-1 3 + tan-1 x = tan-1 8, then x = ____________.
`"sin"^-1 (-1/2)`
`"tan"^-1 (sqrt3)`
If 6sin-1 (x2 – 6x + 8.5) = `pi`, then x is equal to ____________.
The value of `"cos"^-1 ("cos" ((33 pi)/5))` is ____________.
`"cos"^-1 ["cos" (2 "cot"^-1 (sqrt2 - 1))] =` ____________.
The range of sin-1 x + cos-1 x + tan-1 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:
If `"sin"^-1("x"^2 - 7"x" + 12) = "n"pi, AA "n" in "I"`, then x = ____________.
`"cos" ["tan"^-1 {"sin" ("cot"^-1 "x")}]` is equal to ____________.
`"cos"^-1 ("cos" ((7pi)/6))` is equal to ____________.
`"tan"^-1 sqrt3 - "sec"^-1 (-2)` is equal to ____________.
If a = `(2sin theta)/(1 + costheta + sintheta)`, then `(1 + sintheta - costheta)/(1 + sintheta)` is
The number of solutions of sin–1x + sin–1(1 – x) = cos–1x is
Domain and Rariges of cos–1 is:-
Assertion (A): The domain of the function sec–12x is `(-∞, - 1/2] ∪ pi/2, ∞)`
Reason (R): sec–1(–2) = `- pi/4`
`lim_(n→∞)tan{sum_(r = 1)^n tan^-1(1/(1 + r + r^2))}` is equal to ______.
cos–1(cos10) is equal to ______.
Derivative of `tan^-1(x/sqrt(1 - x^2))` with respect sin–1(3x – 4x3) is ______.
`sin[π/3 + sin^-1 (1/2)]` is equal to ______.
Prove that:
tan–1x + tan–1y = `π + tan^-1((x + y)/(1 - xy))`, provided x > 0, y > 0, xy > 1
