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Mathematics Dictionary
Dr. K. G. Shih
Relation Between Functions
Subjects


  • TR 05 00 | - Outlines
  • TR 05 01 | - Pythagorean relations
  • TR 05 02 | - Unit circle
  • TR 05 03 | - Unit hyperbola
  • TR 05 04 | - cos(A)^4 + sin(A)^4 = 1 - 2*(sin(A)^2)*(cos(A)^2)
  • TR 05 05 | - If tan(A) = 2, find (cos(A) + sin(A))^2
  • TR 05 06 | - (cos(A) + sin(A))*(1 - cos(A)*sin(A)) = cos(A)^3 + sin(A)^3
  • TR 05 07 | -
  • TR 05 08 | -
  • TR 05 09 | -
  • TR 05 10 | -
    • Answers


    TR 05 01. Pythagorean relations

    Relations
    • 1. cos(A)^2 + sin(A)^2 = 1.
    • 2. tan(A)^2 + 1 = sec(A)^2.
    • 3. cot(A)^2 + 1 = csc(A)^2.
    Prove that cos(A)^2 + sin(A)^2 = 1
    • cos(A)^2 + sin(A)^2 = (Adj/Hyp)^2 + (Opp/Hyp)^2 = (Adj^2 + Opp^2)/Hyp^2.
    • Since Adj^2 + Opp^2 = Hyp^2.
    • Hence cos(A)^2 + sin(A)^2 = 1.
    Prove that tan(A)^2 + 1 = sec(A)^2
    • cos(A)^2 + sin(A)^2 = 1.
    • Divide both sides by cos(A)^2.
    • Hence 1 + sin(A)^2/cos(A)^2 = 1/cos(A)^2.
    • Hence 1 + tan(A)^2 = sec(A)^2.
    Prove that cot(A)^2 + 1 = csc(A)^2
    • cos(A)^2 + sin(A)^2 = 1.
    • Divide both sides by sin(A)^2.
    • Hence cos(A)^2/sin(A)^2 = 1/sin(A)^2.
    • Hence Cot(A)^2 + 1 = css(A)^2.

    Go to Begin

    TR 05 02. Unit circle

    Prove that cos(A)^2 + sin(A)^2 = 1 give a unit circle.
    • Since x = r*cos(A) and y = r*sin(A)
    • Hence cos(A)^2 + sin(A)^2 = (x/r)^2 + (y/r)^2 = 1
    • Hence x^2 + y^2 = r^2.
    • Since r = 1. Hence it is a circle with radius 1 and center at (0,0).

    Go to Begin

    TR 05 03. Unit hyperbola

    Prove that tan(A)^2 + 1 = sec(A)^2 gives a unit huperbola.
    [Method 1]
    • Since x = tan(A) and y = sec(A)
    • Hence x^2 + 1 = y^2 or x^2 - y^2 = -1.
    • Hence it is a hyperbola with principal axis being y = 0 and sem-axis = 1.
    [Method 2]
    • Let x = sec(A) and y = tan(A)
    • Hence y^2 + 1 = x^2 or x^2 - y^2 = 1.
    • Hence it is a hyperbola with principal axis being x = 0 and sem-axis = 1.

    Go to Begin

    TR 05 04. Prove that cos(x)^4 + sin(x)^4 = 1 - 2*(sin(x)^2)*(cos(x)^2)

    Keyword
    • (a + b)^2 = a^2 + 2*a*b + b^2
    Proof
    • cos(x)^4 + sin(x)^4 = (cos(x)^2 + sin(x)^2)^2 - 2*(sin(x)^2)*(cos(x)^2)
    • = 1 - 2*(sin(x)^2)*(cos(x)^2)

    Go to Begin

    TR 05 05. If tan(A) = 2, find (cos(A) + sin(A))^2

    Use relations of function
    • (cos(A) + sin(A))^2 = cos(A)^2 + sin(A)^2 + 2*sin(A)*cos(A)
    • = 1 + 2*sin(A)*cos(A)*cos(A)/cos(A)
    • = 1 + 2*tan(A)*cos(A)^2
    • = 1 + 2*tan(A)*(1/sec(A)^2)
    • = 1 + 2*Tan(A)/(1 + tan(A)^2)
    • = 1 + 2*2/(1 + 2^2)
    • = 1 + 4/5
    • = 9/5
    Triangle method
    • Since tan(A) = Opp/Adj = 2
    • Hence Opp = 2 and Adj = 1
    • Hence Hyp = Sqr(Opp^2 + Adj^2) = Sqr(2^2 + 1^2) = Sqr(5)
    • Hence sin(A) = Opp/Hyp = 2/Sqr(5) and cos(A) = Adj/Hyp = 1/Sqr(5)
    • Hence (sin(A) + cos(A))^2 = (2/Sqr(5) + 1/Sqr(5))^2
    • = (3/Sqr(5))^2
    • = 9/5

    Go to Begin

    TR 05 06. (cos(A) + sin(A))*(1 - cos(A)*sin(A)) = cos(A)^3 + sin(A)^3

    Keywords
    • (a^3 + b^3) = (a + b)*(a^2 -a*b + b^2)
    Proof
    • (cos(A) + sin(A))*(1 - cos(A)*sin(A))
    • = (cos(A) + sin(A))*(cos(A)^2 + sin(A)^2 - cos(A)*sin(A))
    • = cos(A)^3 + sin(A)^3

    Go to Begin

    TR 05 07. Answer

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    TR 05 08. Answer

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    TR 05 09. Answer

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    TR 05 10. Answer

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    TR 05 00. Outline

    Pythagorean Relations
    • cos(A)^2 + sin(A)^2 = 1
    • tan(A)^2 + 1 = sec(A)^2
    • cot(A)^2 + 1 = csc(A)^2
    Reciprocal Relation
    • sin(x) = 1/csc(x)
    • cos(x) = 1/sec(x)
    • tan(x) = 1/cot(x)
    Unit circle and hyperbola
    • cos(A)^2 + sin(A)^2 = +1 and x^2 + y^2 = +1
    • tan(A)^2 - sec(A)^2 = -1 and x^2 - y^2 = -1
    • sec(A)^2 - tan(A)^2 = +1 and x^2 - y^2 = +1

    Go to Begin
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