Counter Examples
Mathematics Dictionary
Dr. K. G. Shih
Keyword : Tan(45) = 1
Contents of Mathematics Dictionary
Contents

Answers


Q01. Prove that Tan(45) = 1
    * Right angle triangle ABC and angle A=45 degrees. C=90 degrees
    * Theory : Opposite side of angle A = Adjacent side.
    * Trigonometric ratio : Tan(A) = Opp/Adj
    * Hence Tan(45) = Opp/Adj = 1

    * Sin(45) = sqr(2)/2
    * Tan(45) = 1
    * Cos(45) = sqr(2)/2
Angle = 45 degrees = pi/4 radians. pi = 3.1416
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Q02. Value of Tan(A) related with Tan(45)
    * Tan(135) = tan(180 - 45) = tan(45) =-1
    * Tan(225) = tan(180 + 45) =-tan(45) = 1
    * Tan(315) = tan(360 - 45) = tan(45) =-1
    * Tan(405) = tan(360 + 45) =-tan(45) = 1
    * Tan(2*n*pi + pi/4) = tan(pi/4)

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Q03. Why tan(315) is negative ?
A3. Since tan(A)=y/x and 315 degrees in 4th quadrant. x=(+) and y=(-)
    * Angle 045 is in 1st quadrant and tan(045) is positive
    * Angle 135 is in 2nd quadrant and tan(135) is negative
    * Angle 225 is in 3rd quadrant and tan(225) is positive
    * Angle 315 is in 4th quadrant and tan(315) is negative


    tan(A) = (-) | tan(A) = (+)
    -------------|--------------
    tan(A) = (+) | tan(A) = (-)
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Q04. Solve tan(A) = 1 for A between -360 and 360
    1. Angle A between -360 and 360
      * Tan(45) = 1 and A = 45
      * Tan(225) = tan(180+45) = tan45 = 1 and A = 225
    2. General solution
      * Principal angle A = 45 for tan(A) = 1
      * Tan(A) = (+) and A = 45 in 1st quadrant and 225 in 3rd quadrant
      * Hence A = 180*k + 045 in 1st and 3rd quadrant
      * Where k=0,1,2,3,4,... k=even in 1st quadrant and k=odd in 3rd quadrant
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Q05. Solve tan(A) = -1 for A between -360 and 360
    1. Angle A between -360 and 360
      * Tan(180-45) = -tan(45) = -1 and A = 135
      * Tan(360-45) = -tan(45) = -1 and A = 315
    2. General solution
      * Principal angle A = 45 for tan(A)=1
      * Tan(A) = (-) and A is in 2nd quadrant or 4th quadrant
      * Hence A = 180*k - 045
      * Where k=0,1,2,3,4,... k=even in 4th quadrant and k=odd in 2nd quadrant
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Q06. Formula
  • tan(-x) = -tan(x). Hence tangent function is odd function.
  • Co-function
  • tan(090 - A) = +tan(A).
  • tan(090 + A) = -tan(A).
  • tan(270 - A) = +tan(A).
  • tan(270 + A) = -tan(A).
  • tan(n*pi+x) = +tan(x). Hence period of tangent function is pi.
    • tan(180+x) = tan(x).
    • tan(360+x) = tan(x).
    • tan(540+x) = tan(x).
  • tan(n*pi-x) = -tan(x).
    • tan(180-x) = -tan(x).
    • tan(360-x) = -tan(x).
    • tan(540-x) = -tan(x).
  • Ratio definition : tan(A) = Opp/Adj.
  • Pythagorean relation : 1+tan(A)^2 = sec(A)^2 : This is unit hyperbola
  • tan(A) and quadrant
    • 1st quadrant : tan(A) = y/x = (+).
    • 2nd quadrant : tan(A) = y/x = (-).
    • 3rd quadrant : tan(A) = y/x = (+).
    • 4th quadrant : tan(A) = y/x = (-).
  • If y = tan(x) then y' = sec(x)^2. This is the derivative.
  • Slope = tan(A) where A is angle making with x-axis.
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Q07. Find tan(89.99999) and tan(90.00001)
    1. Tan(89.99999) = positive infinite
    2. Tan(90.00001) = negative infinite

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Q08. y = tan(x) and x = arctan(x)
  • If y = tan(x) then the inverse of thangent function is x = arctan(y)
  • Composite function properteis
    • tan(arctan(x) = x.
    • arctan(tan(A) = A.
Solve arctan(-1) = x.
  • By defintion tan(x) = -1.
  • Hence x = pi - pi/4. or x = 2*pi - pi/4.
  • Gerenal solution : x = n*pi - pi/4.
Solve arctan(+1) = x.
  • By defintion tan(x) = 1.
  • Hence x = pi/4. or x = pi + pi/4.
  • Gerenal solution : x = n*pi + pi/4.
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Q09. Special values
  • Angles 0, 90, 180, 270, 360
    • tan(0) = 0
    • Tan(90) = +infinite if angle in 1st quadrant e.g. 89.999999.
    • Tan(90) = -infinite if angle in 2nd quadrant e.g. 90.000001.
    • tan(180) = 0.
    • Tan(270) = +infinite if angle in 3rd quadrant e.g. 269.999999.
    • Tan(270) = -infinite if angle in 4th quadrant e.g. 270.000001.
    • Tan(360) = 0.
  • Angles 30, 45, 60, 120, 135, 150, 210, 225, 240, 300, 315, 330
    • Tan(30) = sqr(3)/3.
    • Tan(45) = 1.
    • Tan(60) = sqr(3).
    • Tan(120) = -sqr(3).
    • Tan(135) = -1.
    • Tan(150) = -sqr(3)/3.
    • Tan(210) = +sqr(3)/3.
    • Tan(225) = +1.
    • Tan(240) = +sqr(3).
    • Tan(300) = -sqr(3).
    • Tan(315) = -1.
    • Tan(330) = -sqr(3)/3.
  • Angles 15, 22.5 and 75 degrees
    • Related with 15 : use tan(A-B) for A = 30 and B = 15.
    • Related with 22.5 : use T(A/2) for A = 45.
    • Related with 76.0 : use Tan(A+B) for A = 30 and B = 45.
  • Angle 9, 18 and 36 degrees.
    • Find the value of sin(18) and cos(18) first.
    • Then use tan(A) = sin(A)/cos(A).
    • Then use tan(2*A) = 2*tan(A)/(1+tan(A)^2).
    • Then use tan(A/2) = Sqr((1-cos(A))/(1+cos(A))).
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Q10. Answer
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Remarks
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