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Mathematics Dictionary
Dr. K. G. Shih
Arcsin(x)


  • Q01 | - Properties of arcsin(x)
  • Q02 | - Find derivative of y = arcsin(x)
  • Q03 | - Series of arcsin(x)
  • Q04 | - Prove that arcsin(x) = arctan(x/Sqr(1 - x^2)
  • Q05 | - arcsin(4/5) + arcsin(5/13) + arcsin(16/65) = pi/2
  • Q06 | - Formula

    • Q01. Properties of arcsin(x)

    Deifintion of y = arcsin(x)
    • If y = arcsin(x) then x = sin(y)
    • Composite function
      • Arcsin(sin(A)) = A.
      • Sin(arcsin(x)) = x.
    • Value of x is between -1 and 1
    Properties of Y=arcsin(x)
      1. Domain : real values of x between -1 an 1
      2. Range : -pi/2 to pi/2
    Special values
    • arcsin(0) = 0
    • arcsin(1/2) = pi/6
    • arcsin(Sqr(2)/2) = pi/4
    • arcsin(Sqr(3)/2) = pi/3
    • arcsin(1) = pi/2

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    Q02. Find derivative of y = arcsin(x)

    • Let y = arcsin(x) and then x = sin(y)
    • Take derivative on both sides with respect x
    • Hence 1 = cos(y)*y'
    • Hence y' = 1/cos(y)
    • Since cos(y)^2 + sin(y)^2 = 1
    • Hence y' = 1/Sqr(1 - sin(y)^2)
    • Hence y' = 1/Sqr(1 - x^2)
    • Note : This the method to change trigonometry to algebra
    Change algebra to trigonometry
    • This is method of the anti-derivative of y = arcsin(x).

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    Q03. Series of arcsin(x)
    Binomial Theory
    • (1 - x^2)^n = 1 + C(n,1)*(x^2) + C(n,2)*(x^2)^2 + ....
    • Let n = -1/2, then
      • C(n, 1) = n = -1/2
      • C(n, 2) = n*(n-1)/2!) = (1/2)*(3/2)/(2!) = 3/8
      • C(n, 3) = n*(n-1)*(n-2)/(3!) = -(1*2*3)/48
      • Etc.
    • 1/Sqr(1 - x^2)
    • = (1 - x^2)^(-1/2)
    • = 1 + (-1/2)*(-x^2) + ((-1/2)*(-1/2 - 1)/(2!))*(x^4) + ....
    • = 1 + (x^2)/2 + (3/8)*x^4 + (1/8)*x^6 + .....
    Series of arcsin(x)
    • Since ∫[1/Sqr(1 - x^2)]dx = arcsin(x)
    • Using binomial theory we have
    • arcsin(x) = ∫[1/Sqr(1 - x^2)]dx
    • = ∫[1 + (x^2)/2 + 3*(x^4)/8 + ....]dx
    • = x - (x^3)/6 + 3*(x^5)/40 + .....
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    Q04. Prove that arcsin(x) = arctan(x/Sqr(1 - x^2)

    arcsin(x) = arctan(x/Sqr(1 - x^2)
    • Draw a right angle triangle
    • Let angle A = arcsin(x)
    • Then Opposite side is x
    • Hypothese is 1
    • Adjacent side = Sqr(1 - x^2)
    • Since tan(A) = Opp/Adj = x/Sqr(1 - x^2)
    • Hence A = arcsin(x) = arctan(x/sqr(1 - x^2))
    arccos(x) = arctan(Sqr(1 - x^2)
    • Draw a right angle triangle
    • Let angle A = arccos(x)
    • Then Adjacent side is x
    • Hypothese is 1
    • Opposite side = Sqr(1 - x^2)
    • Since tan(A) = Opp/Adj = Sqr(1 - x^2)/1
    • Hence A = arccos(x) = arctan(sqr(1 - x^2))

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    Q05. arcsin(4/5) + arcsin(5/13) + arcsin(16/65) = pi/2

    Solution
    Go to Begin

    Q06. Formula

    • sin(A) = Opp/Adj
    • sin(A)^2 + cos(A)^2 = 1
    • y = sin(x) and y' = cos(x)
    • y = arcsin(x) and y' = 1/Sqr(1 - x^2)
    • ∫[1/Sqr(1 - x^2)]dx = arcsin(x)

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