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Mathematics Dictionary
Dr. K. G. Shih
Senior High Mathematics
Study Tips and Home Works
How to do the home work ?

  • 1. Answr the questions first in each home work.
  • 2. Then read the related keywords of the given title in SM.
  • 3. Do the questions again as home work.


  • S41 | - 41. Determinant of order 4
  • S42 | - 42. Binomial Theory : Expand Sqr(1 + x^2) into series
  • S43 | - 43. Binomial Theory : Condition for constant term in (x + 1/(x^2))^n
  • S44 | - 44. Series : Prove that Sum[n*(n+1)/2] = n*(n+1)*(n+2)/6
  • S45 | - 45. Triangle : Construct a triangle using three given medians
  • S46 | - 46. Triangle : Construct a triangle using three given heights
  • S47 | - 47. Ortho-center, centroid and circum-center are conlinear
  • S48 | - 48. Ortho-center : Distance between ortho-center and circum-center
  • S49 | - 49. GP : Angles of triangle in GP terms and common ratio = 3
  • S50 | - 50. Median of triangle ABC : AC^2 + BC^2 = 2*(CF)^2 + (AB^2)/2
  • S51 | - 51. An integer n is expressed as the sum of 3 integers
  • S52 | - 52. Binomila expansion : Prove that 5^(2*n) - 24*n - 1 is divisible by 576
  • S53 | - 53. arcsin(sin(A)+sin(B))+arcsin(sin(A)-sin(B))=pi/2, find sin(A)^2 + sin(B)^2
  • S54 | - 54. S(n) = cos(x) + cos(3*x) + .... + cos((2*n-1)*x)
  • S55 | - 55. S(n) = sin(x)^2 + sin(2*x)^2 + ... + sin(n*x)^2
  • S56 | - 56. y = sin(x) + Sqr(3)*cos(x), find maximum
  • S57 | - 57. sin(A)+sin(B)=p and cos(A)+cos(B)=q, find sin(A+B) and cos(A+B)
  • S58 | - 58. sin(A) and cos(A) have AP mid term sin(x) and GP mid term sin(y), ..
  • S59 | - 59. Find sin(20)*sin(40)*sin(60)*sin(80) without calculator
  • S60 | - 60.
    • Home Works


    S41. Determinant of order 4

    Question : Find x
    • | 1 0 0 0 | = 0
    • | 0 1 0 0 |
    • | 0 0 x 2 |
    • | 0 0 8 x |
    Reference
    Go to Begin

    S42. Binomial Theory : Expand Sqr(1 + x^2)

    Reference     Question
    • Use this series to find Sqr(2)
    Go to Begin

    S43. Binomial Theory : Condition for constant term in (x + 1/(x^2))^n

    Reference     Question
    • What is value of n if there is a constant term

    Go to Begin

    S44. Series : Prove that Sum[n*(n+1)/2] = n*(n+1)*(n+2)/6

    Question
    • 1. Using formula Sum[n] and Sum[n^2}
    • 2. Using Induction (See AL 14 00)

    Go to Begin

    S45. Triangle : Construct a triangle using three given medians

    Reference     Question
    • Given 3 medians of a triangle
    • Construct a triangle

    Go to Begin

    S46. Triangle : Construct a triangle using three given heights

    Reference     Question
    • Given 3 heights of a triangle
    • Construct a triangle

    Go to Begin

    S47. Ortho-center, centroid and circum-center are colinear

    Construction
    • Draw a large triangle ABC
    • Draw ortho-center O
    • Draw centroid G
    • Draw circum-center V
    • Proove that O, G and V are colinear
    • Reference : GE 16 12
    Question
    • Proof by construction and measurement
    • Proof using coordinate geometry
      • Give coordiante of triangle ABC
      • Find coordinate O
      • Find coordinate G
      • Find coordinate V
      • Find slope OG and slope GV
    Reference
    Go to Begin

    S48. Ortho-center : Distance between ortho-center and circum-center

    Construction
    • Draw a large triangle ABC
    • Draw ortho-center O
    • Draw circum-center P
    • Find OP
    • Reference : GE 16 09
    Question 1 : Option
    • Prove that OP^2 = (R^2)*(1 - 8*cos(A)*cos(B)*cos(C))
    • Where R is the circum-radius : a = 2*R*sin(A)
    Question 2 : Use above formula
    • 1. Find OP for a right triangle ABC
    • 2. Find OP for an equilateral triangle
    • 3. Draw a large triangle ABC to find OP. Then very above formula
    Reference
    Go to Begin

    S49. GP : Angles of triangle in GP terms and common ratio = 3

    Question
    • Angles of triangle are in consecutive terms and common ratio = 3
    • Prove that cos(A)*cos(B) + cos(B)*cos(C) + cos(C)*cos(A) = -1/4
    Method and keywords : GP and product of function to sum in trig
    • method 1. Use product of function to sum of function
    • Method 2. Use numerical calculation
    Reference
    Go to Begin

    S50. Median and sides of triangle ABC : AC^2 + BC^2 = 2*(CF^2) + (AB^2)/2

    Constructin
    • Draw a triangle ABC. Let angle CAB less than 90 and CBA greater than 90
    • Let F be mid-point of AB and CF is the median
    Keywords and reference : Median and Pythagorean law
    Go to Begin

    S51. An integer n is expressed as the sum of 3 integers

    Question
    • An integer n is expressed as the sum of 3 integers
    • How many different arrangement ?
    • 1 + 2 + 3 = 6 and 2 + 1 + 3 = 6 are considered as different arrangement
    Keywords and reference : Arrangement
    Go to Begin

    S52. Binomila expansion : Prove that 5^(2*n) - 24*n - 1 is divisible by 576

    Question
    • 5^(2*n) - 24*n - 1 is divisibile by 576
    Keywords and reference : Binomial expansion coefficients
    Go to Begin

    S53. arcsin(sin(A)+sin(B))+arcsin(sin(A)-sin(B))=pi/2, find sin(A)^2 + sin(B)^2

    Keywords
    • If arcsin(u) = v then sin(v) = u
    Reference : Inverse trig function
    Go to Begin

    S54. Series : S(n) = cos(x) + cos(3*x) + cos(5*x) + .... + cos((2*n-1)*x) = ?

    Keywords
    • Product to sum of trig functions
    Question
    • 1. Find S(n)
    • 2. Verify by induction method
    Reference
    Go to Begin

    S55. Series : S(n) = sin(x)^2 + sin(2*x)^2 + ... + sin(n*x)^2

    Keywords
    • sin(x)^2 = (1 - cos(2*x))/2
    • Product to sum of trig functions
    Question
    • 1. Find S(n)
    • 2. Verify by induction method
    Reference
    Go to Begin

    S56 : Find maximum and minimum of y = sin(x) + Sqr(x)

    Keywords
    • sin(A+B) = sin(A)*cos(B) + cos(A)*sin(B)
    • Sin(A) is LE 1 and GE -1
    Questions : Find maximum and minimum
    • using trigonometric method
    • using deriative method
    Reference
    Go to Begin

    S57 : sin(A)+sin(B)=p and cos(A)+cos(B)=q, find sin(A+B) and cos(A+B)

    Keywords
    • Sum to product of trig functions
    • tan(2*x) = 2*tan(x)/(1-tan(x)^2)
    Questions : Find maximum and minimum
    • Find sin(A+B) and cos(A+B)
    Reference
    Go to Begin

    S58 : sin(A) and cos(A) have AP mid term sin(x) and GP mid term sin(y), ...

    Keywords
    • a,b,c in AP : b = (a+c)/2
    • a,b,c in GP : b = Sqr(a*c)
    • sin(A)^2 = 1 - cos(2*A))/2
    Questions :
    • Prove that 2*cos(2*x) = cos(2*y)
    Reference
    Go to Begin

    S59 : Find sin(20)*sin(40)*sin(60)*sin(80) without calculator

    Keywords
  • sin(60) = Sqr(3)/2
  • sin(20)*sin(40) = (cos(40-20)-cos(40+20))/2 Reference
    Go to Begin

    S60 : Find cos(20)*cos(40)*cos(60)*cos(80) without calculator



    Keywords
  • cos(60) = 1/2
  • cos(20)*cos(40) = (cos(40-20)+cos(40+20))/2 Reference

    Go to Begin
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