Mathematics Dictionary

n is integer : (r*(cos(A) + i*sin(A)))^n = (r^n)*(cos(n*A) + i*sin(n*A))
n is negtive integer
- (r*(cos(A) + sin(A)))^n = (r^n)*(cos(2*k*pi+A)/n) + i*sin(2*k*pi+A)/n)
- Where k = 0, 1, 2, ... (n-1)

Diference of angles

1. sin(x-y) = sin(x)*cos(y) - cos(x)*sin(y).
2. cos(x-y) = cos(x)*cos(y) + sin(x)*(sin(y).
3. tan(x-y) = (tan(x) - tan(y))/(1 + tan(x)*tan(y)).

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*** E

Ellipse : Polar form

R = D*e/(1-e*sin(A)) : Focus at bottom as origin
R = D*e/(1+e*sin(A)) : Focus at top as origin
R = D*e/(1-e*cos(A)) : Focus at left as origin
R = D*e/(1+e*cos(A)) : Focus at right as origin

Ellipse : (x/a)^2 + (y/b)^2 = 1 or ((x-h)/a)^2 + ((y-k)/b)^2 = 1

Center : (0,0) or (h,k)
Principal axis
- y = 0 if a GT b
- x = 0 if a LT b
Vertices : (h-a,k) and (h,h+a)
Focal length f = Sqr(a^2 - b^2)
Distance from one focus to directrix = D
Focus : (-f,0) and (f,0)
Equation of directrix : ?
Locus : sum of distance from point P to 2 fixed points equals 2*a

Ellipse : Parametric equation

x = h + a*cos(x) and y = k + b*sin(x)

Equation theory

Quadratic equation a*x^2 + b*x + c = 0 : Roots r1,r2
- Sum of roots = Coefficient of x : r1 + r2 = -b/a or 1/r1 + 1/r2 = -b/c
- Product of roots = Constant term : r1*r2 = c/a
Cubic equation a*x^3 + b*x^2 + c*x + d = 0 : Roots r1,r2,r3
- Sum of roots = coeff of x^2 : r1 + r2 + r3 = -b/a
- Coeff of x : r1*r2 + r1*r3 + r2*r3 = c/a
- Constant term : r1*r2*r3 = -d/a

Ex-central triangle JKL

Side : KJ = 4*R*cos(A/2). (Note A on LK)
Side : LK = 4*R*cos(B/2). (Note B on JL)
Side : JL = 4*R*cos(C/2). (Note C on KJ)
Coordinate of ex-center J (s, s*tan(A/2))
Area of JKL = 8*(R^2)*cos(A/2)*cos(B/2)*cos(C/2)
Circum-radius of JKL = 2*(circum-radius of ABC) = 2*R

Ex-circle

Ex-rdius r = s*tan(A/2)

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*** F

Factor theory : If F(a) = 0, (x-a) is a factor of F(x)

Function : (pi - A) and (pi + A)

sin(pi - A) = +sin(A) and sin(pi + A) = -sin(A)
cos(pi - A) = -cos(A) and cos(pi + A) = -cos(A)
tan(pi - A) = -tan(A) and tan(pi + A) = +tan(A)

Function : (2*pi - A) and (2*pi + A)

sin(2*pi - A) = -sin(A) and sin(2*pi + A) = +sin(A)
cos(2*pi - A) = +cos(A) and cos(2*pi + A) = +cos(A)
tan(2*pi - A) = -tan(A) and tan(2*pi + A) = +tan(A)

Function : Negative angle A

sin(-A) = -sin(A), odd function
cos(-A) = +cos(A), even function
tan(-A) = -tan(A), odd function

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Q07. ***G

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*** H

Half angle

cos(x/2) = Sqr((1 + cos(x))/2)
sin(x/2) = Sqr((1 - cos(x))/2)
tan(x) = Sqr((1 - cos(x))/(1 + cos(x)))

Half angles

sin(A/2) = Sqr((s-b)*(s-c)/(b*c)).
sin(B/2) = Sqr((s-c)*(s-a)/(c*a)).
sin(C/2) = Sqr((s-a)*(s-b)/(a*b)).

Half angles

cos(A/2) = Sqr(s*(s-a)/(b*c)).
cos(B/2) = Sqr(s*(s-b)/(c*a)).
cos(C/2) = Sqr(s*(s-c)/(a*b)).

Half angles

tan(A/2) = Sqr((s-b)*(s-c)/(s*(s-a)).
tan(B/2) = Sqr((s-c)*(s-a)/(s*(s-b)).
tan(C/2) = Sqr((s-a)*(s-b)/(s*(s-c)).

Heron formula : Area of triangle = Sqr(s*(s-a)*(s-b)*(s-c))/(b*c)

Hyperbola : Polar form

R = D*e/(1-e*sin(A)) : Focus at bottom as origin
R = D*e/(1+e*sin(A)) : Focus at top as origin
R = D*e/(1-e*cos(A)) : Focus at left as origin
R = D*e/(1+e*cos(A)) : Focus at right as origin

Hyperbola : (x/a)^2 - (y/b)^2 = 1 or ((x-h)/a)^2 - ((y-k)/b)^2 = 1

Center : (0,0) or (h,k)
Principal axis
- y = k if a GT b
- x = h if a LT b
Vertices : (h-a,k) and (h,h+a)
Focal length f = Sqr(a^2 + b^2)
Distance from one focus to directrix = D
Focus : (-f,0) and (f,0)
Equation of directrix : ?
Locus : difference of distance from point P to 2 fixed points equals 2*a

Hyperbola : Parametric equation

x = h + a*sec(x) and y = k + b*tan(x)

Hyperbolic function

sinh(x) = ((e^x) - e^(-x))/2
cosh(x) = ((e^x) + e^(-x))/2
tanh(x) = sinh(x)/cosh(x)
csch(x) = 1/sinh(x)
tanh(x) = 1/cosh(x)
tanh(x) = cosh(x)/sinh(x)

Hypeebolic function : Inverse (see inverse hyperbolic function)
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*** I

In-circle

in-rdius r = (s - a)*tan(A/2)

Inverse functions

1. arcsin(sin(A)) = A
2. arccos(sin(A)) = A
3. arctan(sin(A)) = A
4. arccsc(sin(A)) = A
5. arcsec(sin(A)) = A
6. arccot(sin(A)) = A

Inverse functions

1. sin(arcsin(x)) = x
2. cos(arccos(x)) = x
3. tan(arctan(x)) = x
4. csc(arccsc(x)) = x
5. sec(arcsec(x)) = x
6. cot(arccot(x)) = x

Inverse hyperbolic functions

(a) arcsinh(X) = ln(X+SQR(X^2+1))
(b) arccosh(X) = ln(X+SQR(X^2-1))
(c) arctanh(X) = ln((1+X)/(1-X))/2
(d) arccsvh(X) = ln(1/X + SQR(X^2+1))
(e) arcsech(X) = ln(1/X + SQR(X^2-1)
(f) arccoth(X) = ln(1-X)/(1+X))/2

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*** J

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Q11. K

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Q12. L

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*** M

Multiple angles

cos(2*x) = 2*cos(x)^2 - 1 = 1 - 2*sin(x)^2
sin(2*x) = 2*sin(x)*cos(x)
tan(2*x) = 2*tan(x)/(1-tan(x)^2)
sin(3*x) = 3*sin(x) - 4*sin(x)^3
cos(3*x) =

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Q14. ***N

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Q15. ***O

Ortho-center : Altitudes of triangle are concurrent
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*** P

Parabola : Polar form

R = D/(1-sin(A)) : Open upward
R = D/(1+sin(A)) : Open downward
R = D/(1-cos(A)) : Open to right
R = D/(1+cos(A)) : Open to left

Parabola : y = a*x^2 + b*x + c

Vertex form : y - k = ((x - h)^2)/(2*D)
Vertex xv = h and yv = k
Distance from focus to directrix = D
Focus : xf = xv and yf = yv + D/2
Equation of directrix : y = yv - D/2
Locus : Point P to fixed point F and fixed line has same distance

Product of functions to sum of functions

sin(A)*sin(B) = (cos(A-B) - cos(A+B))/2.
cos(A)*cos(B) = (cos(A-B) + cos(A+B))/2.
sin(A)*cos(B) = (sin(A+B) - sin(A-B))/2.
cos(A)*sin(A) = (cos(A+B) + cos(A-B))/2.

Pythagorean law : Right triangle ABC c^2 = a^2 + b^2

Pythagorean relations

cos(x)^2 + sin(x)^2 = 1
tan(x)^2 + 1 = sec(x)^2
cot(x)^2 + 1 = csc(x)^2

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*** Q

Quadratic equation : y = F(x) = a*x^2 + b*x + c

Discriminant : D = b^2 - 4*a*c
Quadratic formula
- x = (-b + Sqr(b^2-4*a*c))/(2*a)
- x = (-b + Sqr(b^2-4*a*c))/(2*a)
Vertex : xv = -b/(2*a) and yv = F(xv) = (b^2 - 4*a*c)/(4*a)
y-intecept is c
Slope : y' = 2*a*x + b
Factor form : a*x^2 + b*x + c = (x - r)*(x - s)
- r + s = -b/a
- r*s = c/a