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Mathematics Dictionary
Dr. K. G. Shih
Sine function


  • Q01 | - Sine function : y = sin(x)
  • Q02 | - Sine Law
  • Q03 | - Half angle : sin(A/2) = Sqr(s - b)*(s - c)/(b*c)
  • Q04 | - Sine function and cosine function
  • Q05 | - Locus of R = sin(A)
  • Q06 | - Solve triangle using sine low
  • Q07 | - sin(A) and csc(A)
  • Q08 | - Formula : sin(A + B) and sin(A - B)
  • Q09 | - Formula : series

    • Q01. Sine function

    Triangle definition
    • sin(A) = Opp/Hyp
    Rectangular coordinate definition
    • sin(A) = y/r
    • r = Sqr(x^2 + y^2)
    • sin(A) = (+) in 1st quadrant
    • sin(A) = (+) in 2nd quadrant
    • sin(A) = (-) in 3rd quadrant
    • sin(A) = (-) in 4th quadrant
    Properties of y = sin(x)
    • It is a periodic function and period = 2*pi
    • Five important points between 0 and 2*pi
    • At x = 000 : y = +0
    • At x = 090 : y = +1
    • At x = 180 : y = +0
    • At x = 270 : y = -1
    • At x = 360 : y = +0
    Sketch sine curve : Five point method
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    Q02. Sine Law

    Sine law.
    • Defintion : Trianle ABC
      • a = R*sin(A)
      • b = R*sin(B)
      • c = R*sin(C)
    • Application : Solve triangle using sine law
    Proof of Sine law
    Go to Begin

    Q03. Half angle : sin(A/2) = Sqr(s - b)*(s - c)/(b*c)

    Proof
     Half angle in terms of cos(A)
    • sin(A) = Sqr((1 - cos(A))/2)

    Go to Begin

    Q04. Cosine function and sine function

    Relation
    • sin(A) = +cos(090 - A)
    • sin(A) = +cos(090 + A)
    • sin(A) = -cos(270 - A)
    • sin(A) = -cos(270 + A)
    Pythagorean relation
    • cos(A)^2 + sin(A)^2 = 1
    • This is a unit circle in parametric equation
      • x = cos(t)
      • y = sin(t)

    Go to Begin

    Q05. Locus of R = sin(A)

    Solution : It is a circle
    Go to Begin

    Q06. Solve triangle using sine law

    Example : if a = 2 and b = 3 and angle A = 30 degrees, find other side c.
    • a/sin(A) = b/sin(B)
    • sin(B) = b*sin(A)/a = 3*sin(30)/2 = 3/4
    • B = arcsin(0.75) = 48.59 degrees
    • C = 180 - A - B = 101.41
    • Hence c = a*sin(C)/sin(A) = 2*sin(101.41)/sin(30) =

    Go to Begin

    Q07. sin(A) and csc(A)

    Relation
    • sin(A) = 1/sin(A)

    Go to Begin

    Q08. sin(A + B) and sin(A - B)

    Formula
    • sin(A + B) = sin(A)*cos(B) + cos(A)*sin(B)
    • sin(A - B) = sin(A)*cos(B) - cos(A)*sin(B)
    Proof and application
    • Trigonomery
      • 1. Prove that sin(A + B) = sin(A)*cos(B) + cos(A)*sin(B)
      • 2. Prove that sin(2*A) = 2*sin(A)*cos(A)
      • 3. Prove that sin(3*A) = 3*sin(A)^3 - 4*sin(A)
    Proof with diagrams Product of sin(A)*sin(B)
    • sin(A)*sin(B) = (cos(A - B) - cos(A + B))/2
    Sum of sin(A) + sin(B)
    • sin(A) + sin(B) = 2*(cos((A + B)/2) - cos((A - B)/2)

    Go to Begin

    Q09. Series of sin(x)

    Series
    • sin(x) = x - (x^3)/3! + (x^5)/5! - .....
    Derivative of y = sin(x)
    • y' = +cos(x)
    • y" = +sin(x)
    e^(i*x)
    • e^(i*x) = cos(x) + i*sin(x)

    Go to Begin
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    Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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