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Mathematics Dictionary
Dr. K. G. Shih
Trigonometrical Ratios of Right Triangle
Subjects

  • TR 0200 | - Outlines
  • TR 02 01 | - Trigonometrical ratios
  • TR 02 02 | - Why Opp/Hyp and Adj/Hyp named as sine and cosine function ?
  • TR 02 03 | - Why Opp/Adj and Adj/Opp named as tangent and cotangent ?
  • TR 02 04 | - Find sin(30) by construction a right triangle
  • TR 02 05 | - Find cos(30) by construction a right triangle
  • TR 02 06 | - Find tan(30) by construction a right triangle
  • TR 02 07 | - Use right angle triangle to prove that sin(90-A) = cos(B)
  • TR 02 08 | - Use right angle triangle to prove that cos(90-A) = sin(B)
  • TR 02 09 | - Use right angle triangle to prove that tan(90-A) = cot(B)
  • TR 02 10 | - Co-functions
  • TR 02 11 | - How to get the name of secant and cosecant functions ?
  • TR 02 12 | - Relations indciated on hexagon
  • TR 02 13 | - Prove the relations using right angle triangle
  • TR 02 14 | - Express other ratios in terms of sin(A)
    • Answers


    TR 02 01. Trigonometrical ratios Diagrams
    • Construction of a right angle triangle.
        * Right triangle ABC and angle C = 90 degrees
        * Opp = opposite side of Angle A = BC.
        * Adj = adjacent side of angle A = AC.
        * Hyp = Hypothese of the triangle = Opp of right angle = AB
    • Trigonometric ratios
        * Sin(A) = Opp/Hyp
        * Tan(A) = Opp/Adj
        * Cos(A) = Adj/Hyp
        * Csc(A) = Hyp/Opp = 1/sin(A)
        * Cot(A) = Adj/Opp = 1/tan(A)
        * Sec(A) = Hyp/Adj = 1/cos(A)

    Go to Begin

    TR 02 02. Why Opp/Hyp and Adj/Hyp are named as sine and cosine function ?

    Construction
    • Draw a circle with diameter AB.
    • On circle draw point C.
    • Draw chord AC and BC.
    Chord AC and chord BC
    • Since angle ACB = 90 degrees.
    • Hence Chord BC = AB*sin(A) or sin(A) = BC/AB = Opp/Hyp.
    • Hence Chord AC = AB*cos(A) or cos(A) = AC/AB = Adj/Hyp.
    • These are the defintion of chord functions defined about 200 B.C.
    • Later date the chord functions re-named as sine and cosine functions.
    Diagrams
    • Diagrams The names of sine and cosine : diagram 01 08

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    TR 02 03. Why Opp/Adj and Adj/Opp named as tangent and cotangent function ?

    Construction
    • Draw a circle with radius OX (X-axis).
    • Draw tangent at X.
    • Draw angle A = angle XOP and P is on the tangent.
    • Draw radius OY (Y-axis).
    • Draw tangent at Y and meet OP at Q.
    Tangent XP and tangent YQ
    • Tangent at X is XP with angle POX.
      • tan(XOP) = XP/OP.
      • XP = OP*tan(XOP).
      • Tantent PX is realated with angle XOP as PX = OX*tan(XOP).
    • Tangent at Y is YQ.
      • cot(YQO) = YQ/YO.
      • Since angle YQO = angle QOX = angle XOP
      • YQ = YO*cot(XOP).
      • Tantent QY is realated with angle XOP as QY = OX*cot(XOP).
    Diagrams
    • Diagrams The names of tangent and cotangent : Diagram 01 08

    Go to Begin

    TR 02 04. Find sin(30) by construction a right triangle

    Construction
    • Draw a right triangle ABC.
    • Let angle C = 90 degrees and angle A = 30 degrees.
    • Let AB = 10 cm.
    • Measure BC.
    • Hence sin(A) = sin(30) = BC/AB.

    Go to Begin

    TR 02 05. Find cos(30) by construction a right triangle

    Construction
    • Draw a right triangle ABC.
    • Let angle C = 90 degrees and angle A = 30 degrees.
    • Let AB = 10 cm.
    • Measure AC.
    • Hence cos(A) = cos(30) = AC/AB.

    Go to Begin

    TR 02 06. Find tan(30) by construction a right triangle

    Construction
    • Draw a right triangle ABC.
    • Let angle C = 90 degrees and angle A = 30 degrees.
    • Let AB = 10 cm.
    • Measure AC and BC.
    • Hence tan(A) = tan(30) = BC/AC.

    Go to Begin

    TR 02 07. Use right triangle ABC to prove that sin(90-A) = cos(A)

    Constaction
    • Draw a right triangle ABC.
    • Let angle C = 90 and AB = Hyp.
    Proof
    • For angle A, Adj = AC and Opp = BC.
    • cos(A) = Adj/Hyp = AC/AB.
    • For angle B, Adj = BC and Opp = AC.
    • sin(B) = Opp/Hyp = AC/AB = cos(A).
    • Since angle A + angle B = 90, hence angle B = (90 - A).
    • Hence sin(90 - A) = cos(A)

    Go to Begin

    TR 02 08. Use right triangle ABC to prove that cos(90-A) = sin(A)

    Constaction
    • Draw a right triangle ABC.
    • Let angle C = 90 and AB = Hyp.
    Proof
    • For angle A, Adj = AC and Opp = BC.
    • sin(A) = Opp/Hyp = BC/AB.
    • For angle B, Adj = BC and Opp = AC.
    • cos(B) = Adj/Hyp = BC/AB = sin(A).
    • Since angle A + angle B = 90, hence angle B = (90 - A).
    • Hence cos(90 - A) = sin(A).

    Go to Begin

    TR 02 09. Use right triangle ABC to prove that tan(90-A) = cot(A)

    Constaction
    • Draw a right triangle ABC.
    • Let angle C = 90 and AB = Hyp.
    Proof
    • For angle A, Adj = AC and Opp = BC.
    • cot(A) = Adj/Opp = AC/BC.
    • For angle B, Adj = BC and Opp = AC.
    • tan(B) = Opp/Adj = AC/BC = cos(A).
    • Since angle A + angle B = 90, hence angle B = (90 - A).
    • Hence tan(90 - A) = cot(A).

    Go to Begin

    TR 02 10. Co-function

    Identities in right angle triangle
    • 1. sin(90-A) = cos(A).
    • 2. cos(90-A) = sin(A).
    • 3. tan(90-A) = cot(A).
    • 4. sin(90-A) = cos(A).
    • 5. cos(90-A) = sin(A).
    • 6. tan(90-A) = cot(A).
    Prove sin(90-A) = cos(A)
    • Draw a right angle triangle ABC. Let angle C = 90 and the angle B = 90 - A
    • Opposite side of angle A is BC and adjacent side of angle A is CA
    • Opposite side of angle B is CA and adjacent side of angle B is BC
    • Sin(B) = Opp/Hyp = BC/AB
    • cos(A) = Adj/Hyp = BC/AB
    • Since angle B = 90 - A
    • Hence sin((90-A) = cos(A)

    Go to Begin

    TR 02 11. How to get the name of secant and cosecant functions?

    Construction
    • Draw a circle with diameter AB.
    • Draw tangent at B.
    • Draw secant at A and meet the tangent at P.
    • Hence AP is secant of the circle.
    • Triangle ABP is right triangle.
    • Hence secant AP = AB*sec(PAB)
    • Secant AP meets circle at C.
    • We can draw 2nd secant from B to C and produce to meet tangent from A at Q
    • Hence QB is a secant
    Diagrams
    • Diagrams The names of secant and cosecant : Diagram 01 08

    Go to Begin

    TR 02 12. Relationship between functions

    Diagrams
    • Diagrams Relations indciated on hexagon : diagram 01 05
    • Construction
      • Draw a regular hexagone
      • Put sin(A) on left top vertex
      • Put Tan(A) on left mid vertex
      • Put sec(A) on left bottom vertex
      • Put cos(A) on right top vertex
      • Put cot(A) on right mid vertex
      • Put csc(A) on right bottom vertex
      Relation
      • Left top vertex to right bottom vertex : sin(A) = 1/csc(A)
      • Left mid vertex to right mid vertex : tan(A) = 1/cot(A)
      • Left bottom vertex to right top vertex : sec(A) = 1/cos(A)
    Example : If sin(A) = x, express other ratios in terms of x
    • Since sin(A) = x = Opp/Hyp, hence Opp = x and Hyp = 1.
    • Using Pythagorean law, we have Adj = Sqr(1 - x^2).
    • other ratios in terms of x are
      • cos(A) = Adj/Hyp = Sqr(1-x^2).
      • tan(A) = Opp/Adj = x/Sqr(1-x^2).
      • csc(A) = Hyp/Opp = 1/x.
      • sec(A) = Hyp/Adj = 1/Sqr(1-x^2).
      • cot(A) = Adj/Opp = Sqr(1-x^2)/x.
    Example : Prove that tan(A) = sin(A)/cos(A)
    1st method
    • LHS = tan(A) = Opp/Adj.
    • Divides numberator and denominator by Hyp.
    • Hence LHS = (Opp/Hyp)/(Adj/Hyp) = sin(A)/cos(A)
    2nd method
    • RHS = (Opp/Hyp)/(Adj/Hyp).
    • RHS = (Opp/Hyp)*(Hyp/Adj).
    • RHS = Opp/Adj.
    • RHS = tan(A).
    Example : Prove that csc(A) = 1/sin(A)
    1st method
    • LHS = csc(A) = Hyp/Opp/.
    • Divides numberator and denominator by Hyp.
    • Hence LHS = (Hyp/Hyp)/(Opp/Hyp) = 1/(Opp/hyp) = 1/sin(A).
    2nd method
    • RHS = 1/(Opp/Hyp).
    • RHS = (1)*(Hyp/Opp).
    • RHS = Hyp/Opp.
    • RHS = csc(A).

    Go to Begin

    TR 02 13. Find the relations from right angle triangle

    Example : Prove that tan(A) = sin(A)/cos(A)
    1st method
    • LHS = tan(A) = Opp/Adj.
    • Divides numberator and denominator by Hyp.
    • Hence LHS = (Opp/Hyp)/(Adj/Hyp) = sin(A)/cos(A)
    2nd method
    • RHS = (Opp/Hyp)/(Adj/Hyp).
    • RHS = (Opp/Hyp)*(Hyp/Adj).
    • RHS = Opp/Adj.
    • RHS = tan(A).
    Example : Prove that csc(A) = 1/sin(A)
    1st method
    • LHS = csc(A) = Hyp/Opp/.
    • Divides numberator and denominator by Hyp.
    • Hence LHS = (Hyp/Hyp)/(Opp/Hyp) = 1/(Opp/hyp) = 1/sin(A).
    2nd method
    • RHS = 1/(Opp/Hyp).
    • RHS = (1)*(Hyp/Opp).
    • RHS = Hyp/Opp.
    • RHS = csc(A).

    Go to Begin

    TR 02 14. Express other ratios in terms of sin(A)

    Example : If sin(A) = x, express other ratios in terms of x
    • Since sin(A) = x = Opp/Hyp, hence Opp = x and Hyp = 1.
    • Using Pythagorean law, we have Adj = Sqr(1 - x^2).
    • other ratios in terms of x are
      • cos(A) = Adj/Hyp = Sqr(1-x^2).
      • tan(A) = Opp/Adj = x/Sqr(1-x^2).
      • csc(A) = Hyp/Opp = 1/x.
      • sec(A) = Hyp/Adj = 1/Sqr(1-x^2).
      • cot(A) = Adj/Opp = Sqr(1-x^2)/x.

    Go to Begin

    TR 02 00. How to get the name of secant and cosecant functions?

    Trigonometric ratio and relationship
    • sin(A) = Opp/Hyp
    • cos(A) = Adj/Hyp
    • tan(A) = Opp/Adj = (sin(A))/(cos(A)
    • csc(A) = Hyp/Opp = 1/sin(A)
    • sec(A) = Hyp/Adj = 1/cos(A)
    • cot(A) = Adj/Opp = 1/tan(A) = (cos(A))/(sin(A))
    How to get the names of the functions
    • Why Opp/Hyp and Adj/Hyp named as sine and cosine function ? (TR 02 02)
    • Why Opp/Adj and Adj/Opp named as tangent and cotangent function ? (TR 02 03)
    • Why Hyp/Adj and Hyp/opp named sa secant and cosecant functions? (TR 02 11)
    Co-functions in 1st quadrant
    • 1. sin(90-A) = cos(A).
    • 2. cos(90-A) = sin(A).
    • 3. tan(90-A) = cot(A).
    • 4. sin(90-A) = cos(A).
    • 5. cos(90-A) = sin(A).
    • 6. tan(90-A) = cot(A).
    Function relationships
    • Indicated on hexagon (TR 02 12)
    • Prove using right angle triangle (TR 02 13)
    • Express other ratios in terms of sin(A) if sin(A) = x (TR 02 14)

    Go to Begin
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