Mathematics Dictionary

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Mathematics Dictionary
Dr. K. G. Shih

Trigonometry

Examples Link to program numbers n1 and n2

Read Symbol Defintion

Example : Sqr(x) = Square root of x

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P |
Q | R | S | T | U | V | W | X | Y | Z |

Absolute operation

TR 27 09 y = cos(x) + sin(x) + Abs(cos(x) - sin(x))
TR 27 10 cos(2*x) = sin(x) + Abs(sin(x))
Tr 27 12 Sketch y = cos(x) - Abs(cos(x))
TR 27 10 cos(2*x) = sin(x) + Abs(sin(x)) : find sin(x) between 0 & 2*pi
TR 27 09 y = cos(x)+sin(x) + Abs(cos(x)-sin(x)) : Sketch

Arithmetic Progression

TR 07 21 sin(A) & cos(A) have AP mid terms sin(x) and GP mid term sin(y)
prove that 2*cos(2*x) = cos(2*y)

TR 08 10 Prove that 2*arctan(1/3) + arctan(1/7) = pi/4

TR 27 01 Bisector of triangle AD = 2*b*c*cos(A/2)/(b + c)
Go to Begin

TR 27 04 Equaliteral triangle ABC inscribed unit circle. P on circle.
Prove that PA^2 + PB^2 + PC^2 = 2*3

FI 02 01 Cosine law : prove that cos(A-B) = cos(A)*cos(B) - sin(A)*sin(B)

TR 04 10 cos(1) + cos(2) + ... + cos(179) + cos(180) = -1

TR 07 24 cos(20)*cos(40)*cos(60)*cos(80)

TR 27 10 cos(2*x) = sin(x) + Abs(sin(x)) : find sin(x) between 0 & 2*pi

TR 07 19 cos(A) + cos(B) + cos(C) = ? if A+B+C = pi

TR 07 19 cos(2*A) + cos(2*B) + cos(2*C) = ? if A+B+C = pi

TR 07 25 cos(pi/16)^4 +cos(3*pi/16)^4 +cos(5*pi/16)^4 +cos(7*pi/16)^4

TR 07 14 cos(x) + cos(3*x) + .... + cos((2*n-1)*x)

TR 14 10 cos(x)^6 + sin(x)^6. Find range if x between 0 and pi/2

TR 14 11 cos(x)^2 -4*cos(x)*sin(x) -3*cos(x)^2. Find range if x between 0 & pi/2
Go to Begin

Q04. ***D

Go to Begin

Equaliteral triangle ABC

TR 27 04 : inscribed unit circle. P on circle. prove PA^2 + PB^2 + PC^2 = 2*3

TR 09 07 Equation : x^2 - 6*x + 16*sin(A) = 0
- 1. Find A if equation has integral roots
- 2. Find A if equation has real roots
- 3. Find A if equation hs complex roots
TR 09 08 : 1/cos(x) + 1/sin(x) = 2*Sqr(2)
TR 09 10 : sin(2*x)*sin(5*x) +sin(4*x)*sin(11*x) +sin(9*x)*sin(24*x) = 0
TR 27 10 : cos(2*x) = sin(x) + Abs(sin(x))
TR 27 07 : Solve simultaneous equations
- sin(x)^3 = sin(y)
- cos(x)^3 = cos(y)
TR 27 08 : Solve simultaneous equations
- 2^(cos(x) + sin(y)) = 0
- 16^(cos(x)^2 + sin(x)^2) = 4

Q06. ***F

Go to Begin

TR 07 21 GP : sin(A) & cos(A) have AP mid terms sin(x) & GP mid term sin(y), ...

TR 07 16 GP : angles of triangle in GP and r = 2, prove that cos(A)*cos(B) + ...

TR 27 13 GP : Sum[((1/2)^n)*cos(n*pi/2))] for n = 1 to infinite

TR 27 13 GP : Sum[((1/2)^n)*cos(n*pi/2))] for n = 1 to infinite
Go to Begin

TR 27 05 Hexagon ABCDEF (regular) inscribed unit circle.

P on circle. Prove that PA^2 +PB^2 +PC^2 +PD^2 +PE^2 +PF^2 = 2*6

Identity of triangle ABC

TR 11 01 sin(2*A) + sin(2*B) + sin(2*C) = ?
TR 11 02 cos(2*A) + cos(2*B) + cos(2*C) = ?
TR 11 03 cos(A)^2 + cos(B)^2 + cos(C)^2 = ?
TR 11 04 sin(A)^2 + sin(B)^2 + sin(C)^2 = ?
TR 11 05 sin(A) + sin(B) + sin(C) = ?
TR 11 06 cos(A) + cos(B) + cos(C) = ?
TR 11 07 cos(A/2)^2 + cos(B/2)^2 + cos(C/2)^2 = ?
TR 11 08 sin(A/2)^2 + sin(B/2)^2 + sin(C/2)^2 = ?

TR 05 08 x is between 0 and 2*pi, find x if sin(x) GT cos(x)

Q10. J

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TR 14 10 Max, min of cos(x)^6 + sin(x)^6. x between 0 and pi/2

TR 14 11 Max, min of cos(x)^2 -4*cos(x)*sin(x) -3*cos(x)^2. x between 0 & pi/2

TR 14 06 Max, min of (sec(A)^2 - tan(A))/(sec(A)^2 + tan(A))

TR 27 01 Median of triangle ABC AD = 2*b*c*cos(A/2)/(b+c)

TR 27 02 Medians of triangle ABC make angle x,y,z with sides

Angles : ADB = x, BEC = y and CFA = y
Prove that a*sin(2*x) + b*sin(2*y) + c*sin(2*z)

TR 07 23 Multiple angle : 5 + 8*cos(x) + 4*cos(2*x) + cos(3*x) GT 0
Go to Begin

AN 11 09 Ortho-center to circum-center of triangle ABC

The distance : d^2 = R^2 - 8*(R^2)*cos(A)*cos(B)*cos(C)

TR 07 11 Product of function to sum

Angles of triangle in 3 consecutive GP terms and common ratio = 3
Prove that cos(A)*cos(B) + cos(B)*cos(C) + cos(C)*cos(A) = -1/4

TR 07 16 Product of function to sum

Angles of triangle in 3 consecutive GPterms and common ratio = 2
Find cos(A)*cos(B) + cos(B)*cos(C) + cos(C)*cos(A) = ?

TR 07 00 Product functions to sum

TR 07 14 : S(n) = cos(x) + cos(3*x) + .... + cos((2*n-1)*x)
TR 07 15 : S(n) = sin(x)^2 + sin(2*x)^2 + .... + sin(n*x)^2

TR 07 14 : S(n) = cos(x) + cos(3*x) + .... + cos((2*n-1)*x)
TR 07 15 : S(n) = sin(x)^2 + sin(2*x)^2 + .... + sin(n*x)^2

TR 07 09 sin(1) + sin(2) + ... + sin(359) + sin(360) = 0

TR 07 24 sin(20)*sin(40)*sin(60)*sin(80)

TR 07 25 sin(pi/16)^4 +sin(3*pi/16)^4 +sin(5*pi/16)^4 +sin(7*pi/16)^4

TR 27 05 Square ABCD inscribed unit circle.

P on circle. Prove that PA^2 +PB^2 +PC^2 +PD^2 = 2*4

Sum to product functions

TR 07 19 : cos(2*A) + cos(2*B) + cos(2*C) = ? if A+B+C = pi
TR 07 19 : sin(A)^2 + sin(B)^2 + sin(C)^2 = ? if A+B+C = pi
TR 07 19 : cos(A) + cos(B) + cos(C) = ? if A+B+C = pi
TR 07 20 : sin(A)+sin(B)=p and cos(A)+cos(B)=q, find sin(A+B)
TR 07 22 : If sin(x+y) = sin(x)+sin(y), find conditions
TR 07 22 : If x and y are acute angles, sin(x+y) LT sin(x)+sin(y)

TR 27 11 (1 + tan(15))/(1 - tan(15)) = ?

TR 10 12 tan(A/2)*tan(B/2) + tan(B/2)*tan(C/2) + tan(C/2)*tan(A/2)

TR 18 03 Transformation : y = sin(x) to y - 2 = 3*Sin(2*x + pi/4)

TR 07 19 : cos(2*A) + cos(2*B) + cos(2*C) = ?
TR 07 19 : sin(A)^2 + sin(B)^2 + sin(C)^2 = ?
TR 07 19 : cos(A) + cos(B) + cos(C) = ?

TR 14 05 x^2 - 2*x*sin(pi*x/2) + 1 = 0 has real roots

TR 09 07 x^2 - 6*x + 16*sin(A) = 0 has integral roots
Go to Begin

TR 27 09 y = cos(x)+sin(x) + Abs(cos(x)-sin(x)) : Sketch

Tr 27 12 y = cos(x) - Abs(cos(x)) : sketch
Go to Begin

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