Advertisements
Advertisements
प्रश्न
Evaluate the following:
`tan^-1 1+cos^-1 (-1/2)+sin^-1(-1/2)`
Advertisements
उत्तर
Let `sin^-1(-1/2)=y`
Then,
`siny=-1/2`
We know that the range of the principal value branch is `[-pi/2,pi/2].`
Thus,
`siny=-1/2=sin(-pi/6)`
`=>y=-pi/6in[-pi/2,pi/2]`
Now,
Let cos^-1(-1/2)= z
Then,
`cosz=-1/2`
We know that the range of the principal value branch is [0, π].
Thus,
`cosz=-1/2=cos((2pi)/3)`
`=>z = (2pi)/3in[0,pi]`
so
`tan^-1 1+cos^-1(-1/2)+sin^-1(1/2)=pi/4+(2pi)/3-pi/6=(3pi)/4`
`therefore tan^-1 1+cos^-1(-1/2)+sin^-1(1/2)=(3pi)/4`
APPEARS IN
संबंधित प्रश्न
Find the principal value of the following:
tan−1 (−1)
Find the principal value of the following:
`sec^(-1) (2/sqrt(3))`
Find the principal value of the following:
`cot^(-1) (sqrt3)`
Find the principal value of the following:
`"cosec"^(-1)(-sqrt2)`
Find the value of the following:
`cos^(-1) (1/2) + 2 sin^(-1)(1/2)`
If sin−1 x = y, then ______.
Find the value of the following:
`cos^(-1) (cos (13pi)/6)`
`sin^-1 1/2-2sin^-1 1/sqrt2`
Find the domain of the following function:
`f(x)sin^-1sqrt(x^2-1)`
Evaluate the following:
`tan^-1(-1/sqrt3)+tan^-1(-sqrt3)+tan^-1(sin(-pi/2))`
Evaluate: tan `[ 2 tan^-1 (1)/(2) – cot^-1 3]`
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 sinA.
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)`
In ΔABC, prove the following:
`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`
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
Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`
Evaluate:
`cos[tan^-1 (3/4)]`
Express `tan^-1 [(cos x)/(1 - sin x)], - pi/2 < x < (3pi)/2` in the simplest form.
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then θ = ______
`sin{tan^-1((1 - x^2)/(2x)) + cos^-1((1 - x^2)/(1 + x^2))}` is equal to ______
`cos^-1 4/5 + tan^-1 3/5` = ______.
The domain of the function defined by f(x) = sin–1x + cosx is ______.
All trigonometric functions have inverse over their respective domains.
When `"x" = "x"/2`, then tan x is ____________.
`"cos"^-1 ["cos" (2 "cot"^-1 (sqrt2 - 1))] =` ____________.
`"cos" ["tan"^-1 {"sin" ("cot"^-1 "x")}]` is equal to ____________.
`2"tan"^-1 ("cos x") = "tan"^-1 (2 "cosec x")`
`"cos"^-1 ("cos" ((7pi)/6))` is equal to ____________.
`"tan"^-1 sqrt3 - "sec"^-1 (-2)` is equal to ____________.
Assertion (A): The domain of the function sec–12x is `(-∞, - 1/2] ∪ pi/2, ∞)`
Reason (R): sec–1(–2) = `- pi/4`
cos–1(cos10) is equal to ______.
If sin–1a + sin–1b + sin–1c = π, then find the value of `asqrt(1 - a^2) + bsqrt(1 - b^2) + csqrt(1 - c^2)`.
The value of `cos^-1(cos(π/2)) + cos^-1(sin((2π)/2))` is ______.
Find the value of `tan^-1(x/y) + tan^-1((y - x)/(y + x))`
