S61 |
- 61. If x,y are acute angle, then sin(x+y) LT (sin(x)+sin(y))
S62 |
- 62. If sin(x+y) = (sin(x)+sin(y)), find the conditions
S63 |
- 63. Show that 5 + 8*cos(x) + 4*cos(2*x) + cos(3*x) GE 0
S64 |
- 64. sin(1*pi/16)^4 +sin(3*pi/16)^4 +sin(5*pi/16)^4 +sin(7*pi/16)^4 = ?
S65 |
- 65. cos(1*pi/16)^4 +cos(3*pi/16)^4 +cos(5*pi/16)^4 +cos(7*pi/16)^4 = ?
S66 |
- 66. Maximum and Minimum : (sec(A)^2 - tan(A))/(sec(A)^2 + tan(A))
S67 |
- 67. sin(1) + sin(2) + ..... + sin(369) + sin(360)
S68 |
- 68. y = sin(x) and R = sin(A) : Comparison
S70 |
- 70. y = sin(x) and y = sinh(x) : comparison
S71 |
- 71. Divide triangle ABC into four equal area triangles
S72 |
- 72. E,F,T on sides of triangle ABC. Area EFT
S73 |
- 73. E,F,T on sides of triangle ABC. Area EFT
S74 |
- 74. E,F,T on sides of triangle ABC. Area EFT
S75 |
- 75. Sequece : 1, 3, 7, 13, 21, ....
S76 |
- 76. Describe curve y = (x^2)/((x^2 - 1)^2)
S77 |
- 77. Prove that graph of R = sin(2*A) has four petals
S78 |
- 78. Petals of R = sin(3*A)
S79 |
- 79. Petals of R = sin(3*A/2)
S80 |
- 80 Solve Simultaneous equations sin(x)^3 = sin(y) and cos(x)^3 = cos(y)
Home Works
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S61. If x,y are acute angle, then sin(x+y) LT (sin(x)+sin(y))
Keyword
- sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y)
Solution : See TR 07 22.
Go to Begin
S62. If sin(x+y) = (sin(x)+sin(y)), find the conditions
Keyword
- sin(x+y) = 2*sin((x+y)/2)*cos((x+y)/2)
- sin(x) + sin(y) = 2*(sin((x+y)/2)*cos((x-y)/2)
Solution
Go to Begin
S63. Show that 5 + 8*cos(x) + 4*cos(2*x) + cos(3*x) GE 0
Keywords
- cos(2*x) = 2*cos(x)^2 - 1
- cos(3*x) = 4*cos(x)^3 - 3*cos(x)
- Use synthetic division to solve a cubic equation
Solution
Go to Begin
S64. sin(1*pi/16)^4 +sin(3*pi/16)^4 +sin(5*pi/16)^4 +sin(7*pi/16)^4 = ?
Question
- Find the value of expression without calculator
- Verify the results using calculator
keywords
- sin(1*pi/16)^4 = ((1 - cos(1*pi/8)/2)^2
- cos(A)+cos(B) = (cos((A+B)/2)*cos((A-B)/2))/2
Solution
Go to Begin
S65. cos(1*pi/16)^4 +cos(3*pi/16)^4 +cos(5*pi/16)^4 +cos(7*pi/16)^4 = ?
Question
- Find the value of expression without calculator
- Verify the results using calculator
keywords
- cos(1*pi/16)^4 = ((1 + cos(1*pi/8)/2)^2
- cos(A)+cos(B) = (cos((A+B)/2)*cos((A-B)/2))/2
Solution
Go to Begin
S66. Maximum and Minimum : (sec(A)^2 - tan(A))/(sec(A)^2 + tan(A))
1 + tan(A)^2 = sec(A)^2
Let y = (sec(A)^2 - tan(A))/(sec(A)^2 + tan(A))
Simplify and get quadratic function of tan(x)
Solution
Go to Begin
S67. sin(1) + sin(2) + sin(3) + ...... + sin(359) + sin(360)
Keyword
- use sin(360 - A) = - sin(A)
- Or use sin(A) + sin(B) = 2*sin((A+B)/2)*cos((A-B)/2)
Solution
Go to Begin
S68. y = sin(x) and R = sin(A) : Comparison
Note
- 1. y = sin(x) is function in rectangular coordinates
- 2. R = sin(A) is function in polar coordiantes
Solution
Go to Begin
S69. Describe the plot of x = sin(t) and y = sin(t)
Question : Describe the plot
- 1. t = 0 to pi/2
- 2. t = pi/2 to pi
- 3. t = pi to 3*pi/2
- 4. t = 3*pi/2 to 2*pi
Question : Describe the final plot in oxy coordinate
Solution
Go to Begin
S70. y = sin(x) and y = sinh(x) : Comparison
Questions
- 1. The difference
- 2. The similarity
- 3. Why does it name sinh(x)
Solution
Go to Begin
S71. Divide triangle ABC into four equal area triangles
Question
- Draw a triangle ABC
- Let mid points be E on CA, F on BC, T on AB
- Join the mid points and give four equal area triangles
- Also proof the area of each triangle is (R^2)*sin(A)>sin(B)*sin(C)
- Where R = circum-radius
Reference : Trigonometric method
Reference : Geometric method
Go to Begin
S72. Tangle ABC has four triangles by joining points on sides
Question
- Draw a triangle ABC and let a, b, c are the sides
- Let F on BC, and CF = BF = a/2
- Let E on CA, and CE = 2*b/3 and EA = b/3
- Let T on AB
- Join the points and give four 4 triangles
- Let y = area of triangle CEF
- Let z = area of triangle EAT
- Let x = area of triangle BFT
- If x^2 = y*z, find BT : TA
Reference : Trigonometric method
Reference : Trigonometric method
Go to Begin
S73. E,F,T on sides of triangle, Area EFT = (area ABC)*(n^2-3*n+3)/(n^2)
Construction
- F on BC and FC = BC/2 = c/n
- E on CA and EA = CA/3 = b/n
- T on AB and TB = AB/n = a/n
Question
- Prove that are EFT = 7*(area ABC)/24
Reference
Go to Begin
S74. E,F,T on sides of triangle, Area EFT = 7*(area ABC)/24
Construction
- F on BC and FC = BC/2 = c/2
- E on CA and EA = CA/3 = b/3
- T on AB and TB = AB/4 = a/4
Question
- Prove that are EFT = 7*(area ABC)/24
Reference
Go to Begin
S75. Sequence : 1, 3, 7, 13, 21, ....
Questions
- 1. T(1) = 1 and T(2) = 3, Find T(n)
- 2. Find S(n)
Reference
Go to Begin
S76 : Decribe the curve y = (x^2)/((x^2 - 1)^2)
Questions
- 1. The asymptotes
- 2. The signs of values of y
- 3. The concavity of the curve
- 4. The range of the curves is increasing
Reference
Go to Begin
S77. Prove that graph of R = sin(2*A) has four petals
Questions
- 1. Find the directions of four petals
- 2. Find the cycle domain
- 3. Compare with R = cos(2*A)
Reference :
Go to Begin
S78. Prove that graph of R = sin(3*A) has three petals
Questions
- 1. Find the directions of three petals
- 2. Find the cycle domain
Reference :
Go to Begin
S79 : Find number of petals of R = sin(3*A/2)
Questions
- 1. Find the directions of six petals
- 2. Find the cycle domain
- 3. Compare with R = cos(3*A/2)
Reference :
Go to Begin
S80. Solve Simultaneous equations sin(x)^3 = sin(y) and cos(x)^3 = cos(y)
Reference
Go to Begin
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