The Arctangent Function Y=arcsin(x)
Examples of arcsin(x) in ZA.txt
Read Instructions
Keywords : Arcsin(x)
Subjects

Q1. Hightlight
  • Defintion of arcsin(x).
  • Properties of y = arcsin(x).
    • sin(arcsin(x) = x.
    • arcsin(sin(x) = x.
  • Express arcsin(x) in terms of arctan(x).
  • Derivative of y = arcsin(x) - Change trigonometry to algebra.
  • Change algebra to trigonometry : The integral of 1/Sqr(1-x^2).
  • Series of arcsin(x).
  • d/dx(arcsin(x)) = 1/Sqr(1-x^2).
  • Integral of (1/Sqr(1-x^2))*dx = arcsin(x).
  • arcsin(1/2) = x
    • x = pi/6 or x = pi - pi/6.
    • General solution is 2*n*pi + pi/6 or 2*(n+1)*pi - pi/6.
Go to Begin

Q2. Deifintion
    1. If y = arcsin(x) then x = sin(y)
    2. Composite function
    • Arcsin(sin(A)) = A
    • Sin(arcsin(x)) = x
Examples
  • Find arcsin(sin(45.75)). Answer : arcsin(sin(45.75)) = 45.75 degrees.
  • Find sin(arcsin(0.255)). Answer : sin(arcsin(0.255)) = 0.255.
Go to Begin

Q3. Properties of Y = arcsin(x)
    1. Domain : x is between -1 and 1
    2. Range : -pi/2 to pi/2
    3. The curve is always increasing since y' = 1/sqr(1-x^2) is positive
    4. Find y".
    • y" = -(d/dx(Sqr(1-x^2)/(1-x^2)) = 2*x/((1-x^2)^(3/2).
    • y" = 0 and at x = 0 hence the point of inflection is at x = 0
    • Concavity :
      • concave downward if x LT 0 since y" LT 0.
      • concave upward if x GT 0 since y" GT 0.
Why y = arcsin(x) and x is between -1 and 1.
  • Since A = arcsin(x) and x = sin(A).
  • The amplitude of sine curve is 1. Hence value of x is between -1 and 1.
  • Maximum sin(90) = 1 and minimum sin(27) = -1.
Go to Begin

Q4. Find derivative
    1. Let y = arcsin(x) and then x = sin(y)
    2. Take derivative on both sides
    3. Hence 1 = cos(y)*y'
    4. Hence y' = 1/cos(y) = 1/sqr(1-sin(y)^2) = 1/sqr(1-x^2)
    5. This the method to change trigonometry to algebra
Example : Change 1/sqr(1-x^2) back to arcsin(x)
  • Integeral of dx/(1-x^2) = arcsin(x) + C.
  • This is anti-dirivative of arcsin(x).
  • This is change algebra to trigonometry.
Go to Begin

Q5. Series of arcsin(x)
  • arcsin(x) = x + x^2/(2*3) + (1*3*x^5)/(2*4*5) + (1*3*5*x^5)/(2*4*6*7) + ...
  • This series is hard to remember
Go to Begin

Q6. Express arcsin(x) in terms of arctan(x)
    1. Let A = arcsin(x)
    2. Sin(A) = x/1 = Opp/Hyp.
    • Hence Opp = x and Hyp = 1.
    • Hence Adj = Sqr(1-x^2)
    3. Tan(A) = Opp/Adj = x/sqr(1-x^2)
    4. Hence A = arctan(x/sqr(1-x^2))
    5. Hence arcsin(x) = arctan(x/sqr(1-x^2))
Example : If arcsin(-1/2) = A, find tan(A)
  • Since arcsin(-1/2) = A, hence sin(A) = -1/2.
  • Hence A is in 3rd and A = 2*(n+1)*pi + pi/6.
  • Hence tan(A) = +tan(pi/6) = sqr(3)/3.
  • Or A is in 4th and A = 2*n*pi - pi/6.
  • Hence tan(A) = -tan(pi/6) = -sqr(3)/3.
Go to Begin

Q7. Formula
  • Trignometric ration
    • sin(A) = Opp/Hyp.
    • cos(A) = Adj/Hyp.
    • tan(A) = Opp/Adj.
  • Pythagorean relation
    • cos(A)^2 + sin(A)^2 = 1. This uint circle.
    • tan(A)^2 + 1 = sec(A)^2. This unit hyperbola.
    • cot(A)^2 + 1 = csc(A)^2. This unit hyperbola.
  • Derivatives
    • If y = sin(x) then y' = cos(x).
    • If y = cos(x) then y' =-sin(x).
    • If y = tan(x) then y' = sec(x)^2.
  • Derivative of arcsin(x)
    • y = arcsin(x) and y' = 1/sqr(1-x^2).
  • Anti-derivative of arcsin(x)
    • ∫[1/sqr(1-x^2)]dx = arcsin(x)
Values of special angle
  • sin(pi/6) = 1/2 and arcsin(1/2) = pi/6.
  • sin(pi/4) = sqr(2)/2 and arcsin(sqr(2)/2) = pi/4.
  • sin(pi/3) = sqr(3)/2 and arcsin(sqr(3)/2) = pi/3.
  • sin(pi/2) = 1 and arcsin(1) = pi/2.
General solution of sin(x) = c
  • If c is positive, then x = 2*(n+1)*pi - A or x = 2*n*pi + A.
  • If c is negative, then x = 2*(n+1)*pi + A or x = 2*n*pi - A.
  • If c is positive and negative, then x = n*pi + A or x = n*pi - A.
  • Where A is the principal of arcsin(c) in 1st quadrant
Go to Begin

Q8. Reference
  • Graphs relation with sine : MD2002 ZM31
  • Graph of y = sin(x) : On internet ABH program 04 01
  • Arcsin in Text file : MD2002 ZA.txt
 Go to Begin
Q09. Solve arcsin(-1/2) = x
  • Since arcsin(-1/2) = x, hence sin(x) = -1/2.
  • The pricipal angle of arcsin(-1/2) is arcsin(1/2) = pi/6.
  • Hence x = 2*n*pi + pi/2 or x = 2*n*pi - pi/2.
  • For answer between 0 and 2*pi is x = 210 or 330 degrees.
Go to Begin

Q10. Quiz for y = arcsin(x)
  • 1. Find arcsin(sin(44 deg)).
  • 2. Find sin(arcsin(0.2345)).
  • 3. Find arcsin(2.5).
  • 4. If arcsin(-Sqr(3)/2) = x, find x between 0 and 360 degrees.
  • 5. If arcsin(-Sqr(2)/2) = x, find x in genearal solutions.
Go to Begin

Q11. Answer to Quiz for y = arcsin(x)
  • 1. arcsin(sin(44 deg)) = 44 degrees.
  • 2. sin(arcsin(0.2345)) = 0.2345.
  • 3. arcsin(2.5) does not exist since for arcsin(x) and x is between -1 and 1.
  • 4. If arcsin(-Sqr(3)/2) = x, then x = 180 + 60 or x = 360 - 60.
  • 5. If arcsin(-Sqr(2)/2) = x, then x = 2*(n+1)*pi + pi/3 or x = 2*n*pi - pi/3.
Go to Begin
Show Room of MD2002 Contact Dr. Shih Math Examples Room
Copyright © Dr. K. G. Shih. Nova Scotia, Canada.
1