Miscellaneous Formulas
Power Sums:
Sum of first n integers = n(n+1)/2
Sum of first n squares = (2n3+3n2+m)/6
Sum of first n cubes = (n4+2n3+n2)/4
Sum of first n 4th powers = (6n5 + 15n4 + 10n3 - n)/30
Sum of first n 5th powers = (2n6 + 6n5 + 5n4 - n2)/12
Algebraic:
Ax2 + Bx + C = 0. x= (-B +/- sqrt(B2-4AC))/(2A)
Series:
1/(1-x) = 1 + x + x2 + x3 + ... [-1 < x < 1]
ln(1+x) = x - x2/2 + x3/3 - x4/4 + ... [-1 < x <= 1]
sin(x) = x - x3/3! + x5/5! - x7/7! + ...
cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...
tan(x) = x + x3/3 + 2x5/15 + 17x7/315 + 62x9/2835+(22n (22n-1)Bnx2n-1)/(2n)! + ... [|x| < Pi/2]
tan-1(x) = x - x3/3 + x5/5 - x7/7 + ... [-1 <= x <= 1]
ex = 1 + x + x2/2! + x3/3! + ...
sec(x) = 1 + x2/2 + 5x4/24 + 61x6/720 + ... + Enx2n/(2n)! + ...
csc(x) = 1/x + x/6 + 7x3/360 + 31x5/15120 + (2(22n-1-1)Bnx2n-1)/(2n)! + ...
cot(x) = 1/x - x/3 - x3/45 - 2x5/945 - (22n Bnx2n-1)/(2n)! - ...
sinh(x) = x + x3/3! + x5/5! + x7/7! + ...
cosh(x) = 1 + x2/2! + x4/4! + x6/6! + ...
Area, Volume, Perimeter:
Area(Triangle) = base*height/2
Area(Triangle), given lengths of sides a,b,c = sqrt(s(s-a)(s-b)(s-c)) where s=(a+b+c)/2. "Heron's Formula"
Area(Circle) = Pi*r2
Perimeter(Circle) = 2*Pi*r
Volume(Cylinder) = Pi*r2*h
Volume(Right Circular Cone) = Pi*r2*h/3
Volume(Right Circular Cone with missing top [Frustrum]) = Pi*(r2 + rR + R2)*h/3. where 'R' is radius of base and 'r' is radius of top.
Volume(Sphere) = 4*Pi*r3/3
Surface Area(Cylinder) = 2*Pi*(r2 + r*h)
Surface Area(Cone) = Pi*(r2 + r*sqrt(r2 + h2))
Surface Area(Sphere) = 4*Pi*r2
Trig:
tan(x) = sin(x)/cos(x)
cot(x) = 1/tan(x) = cos(x)/sin(x)
sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
sin2(x) + cos2(x) = 1
1 + tan2(x) + sec2(x)
1 + cot2(x) + csc2(x)
sin(Pi/2-x) = cos(x)
cos(Pi/2-x) = sin(x)
tan(Pi/2-x) = cot(x)
sin(-x) = -sin(x)
cos(-x) = cos(x)
tan(-x) = -tan(x)
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
tan(x+y) = (tan(x)+tan(y)) / (1 - tan(x)tan(y))
sin(x-y) = sin(x)cos(y) - cos(x)sin(y)
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
tan(x-y) = (tan(x)-tan(y)) / (1 + tan(x)tan(y))
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos2(x) - sin2(x) = 1 - 2sin2(x) = 2cos2(x)-1
tan(2x) = 2tan(x)/(1-tan2(x))
tan(x/2) = (1-cos(x))/sin(x)
sin(x/2) = +/- sqrt((1-cos(x))/2)
cos(x/2) = +/- sqrt((1+cos(x))/2)
sinh(x) = (ex - e-x)/2
cosh(x) = (ex + e-x)/2
acos(n*i) = Pi/2 - i*ln(n + sqrt(n2+1))
acos(n*i) = Pi/2 - i*sinh-1(n)
sin2(x) =(1 - cos(2x))/2
cos2(x) =(1 + cos(2x))/2
tan2(x) =(1 - cos(2x))/(1 + cos(2x))
sin(x) + sin(y) = 2 sin((x+y)/2) cos((x-y)/2)
sin(x) - sin(y) = 2 cos((x+y)/2) sin((x-y)/2)
cos(x) + cos(y) = 2 cos((x+y)/2) cos((x-y)/2)
cos(x) - cos(y) = -2 sin((x+y)/2) sin((x-y)/2)
sinh(2 sinh-1(n)) = 2n*sqrt(n2+1)
sin(tan-1(x)) = x/sqrt(x2+1)
cos(tan-1(x)) = 1/sqrt(x2+1)
tan(sin-1(x)) = x/sqrt(1-x2)
tan(cos-1(x)) = sqrt(1-x2)/x
tan(0.5*cos-1(x)) = i*sqrt((x-1)/(x+1))
cos(sin-1(x)) = sin(cos-1(x)) = i*sqrt(x2-1)
Here's one I did by hand back in high school (radius of inscribed circle)...
In a triangle with sides a,b,c: Radius of inscribed circle =
(a+b-c) * tan(0.5*acos( (a2+b2-c2)/(2ab) )) / 2
Radius of inscribed circle (without trig): (a+b-c)*sqrt((2ab+c2-a2-b2) / (2ab-c2+a2+b2)) / 2
Radius of inscribed circle (easy): r = Area / Semiperimeter. Note: Semiperimeter = (a+b+c)/2
Center of inscribed circle is where angle bisector lines intersect.
Radius of circumscribed circle is a*b*c/(4*Area).
Center of circumscribed circle is where the perpendicular-bisectors of the triangle sides intersect.
Financial:
Loan Payment = p*(i/12) * (1+i/12)^m / ((1+i/12)^m - 1) where 'm' is number of monthly payments, 'i' is the annual interest rate, and 'p' is amount borrowed.
Compound Interest = (1+i/n)^(n*y) where 'i' is the interest rate, 'n' is the number of times per year compounding is done, and 'y' is number of years to compound. Multiply result by principal to determine principal and interest.
Derivatives:
d/dx k*f(x) = k * d/dx f(x)
d/dx |x| = |x|/x
d/dx xr = rxr-1
d/dx sin(x) = cos(x)
d/dx cos(x) = -sin(x)
d/dx tan(x) = sec2(x)
d/dx cot(x) = -csc2(x)
d/dx sec(x) = sec(x) tan(x)
d/dx csc(x) = -csc(x) cot(x)
d/dx sinh(x) = cosh(x)
d/dx coth(x) = -csch2(x)
d/dx cosh(x) = sinh(x)
d/dx sech(x) = -sech(x) tanh(x)
d/dx tanh(x) = sech2(x)
d/dx csch(x) = -csch(x) coth(x)
d/dx ln(x) = 1/x
d/dx ex = ex
d/dx log_a(x) = 1/(x ln(a))
d/dx ax = ax ln(a)
d/dx sin-1(x) = 1/sqrt(1-x2)
d/dx cos-1(x) = -1/sqrt(1-x2)
d/dx tan-1(x) = 1/sqrt(1+x2)
d/dx sec-1(x) = 1/(|x| sqrt(x2-1))
Integrals:
Int[u] dv = uv - Int[v] du
Int[un] du = un+1/(n+1) + C (n != 1)
Int[1/u] du = ln |u| + C
Int[eu] du = eu + C
Int[au] du = au/ln(a) + C
Int[sin(u)] du = -cos(u) + C
Int[cos(u)] du = sin(u) + C
Int[sec2(u)] du = tan(u) + C
Int[csc2(u)] du = -cot(u) + C
Int[sec(u) tan(u)] du = sec(u) + C
Int[csc(u) cot(u)] du = -csc(u) + C
Int[tan(u)] du = ln|sec(u)| + C
Int[cot(u)] du = ln|sin(u)| + C
Int[sec(u)] du = ln|sec(u) + tan(u)| + C
Int[csc(u)] du = ln|csc(u) - cot(u)| + C
Int[1/sqrt(a2-u2)] du = sin-1(u/a) + C
Int[1/(a2+u2)] du = (1/a)tan-1(u/a) + C
Int[1/(a2-u2)] du = (1/(2a)) ln|(u+a)/(u-a)| + C
Int[1/(u*sqrt(u2-a2)] du = (1/a)sec-1(u/a) + C
Conversions:
1 inch = 2.54 centimeters
1 liter = 1000 cubic centimeters
1 liter = 1.057 quarts
1 kilogram = 2.20 pounds
1 kilometer = 0.62 miles
1 pound = 453.6 grams
Pi radians = 180 degrees
Special Numbers:
Bernoulli Polynomials and Numbers: x*ex*t/(ex-1) = Sum from n=0 to infinity, of [Cn(t) xn/(n!)] where |x| < 2Pi
When t=0 and Bn=(-1)n-1 C2n(0), the above expression becomes:
x/(ex-1) = 1 - x/2 + B1x2/2! - B2x4/4! + B3x6/6! - ...
B1=1/6, B2=1/30, B3=1/42, B4=1/30, B5=5/66, B6=691/2730, B7=7/6, B8=3617/510, B9=43867/798, ...
Euler Polynomials and Numbers: 2*ex*t/(ex+1) = Sum from n=0 to infinity, of Dn(t) xn/(n!) where|x| < Pi
When t=1/2 and En=(-1)n 22n D2n(1/2), the above expression becomes:
2*(sqrt(e))x/(ex+1) = 1 - E1 x2/(22*2!) + E2 x4/(24*4!) + E3 x6/(26*6!) - ...
E1=1, E2=5, E3=61, E4=1385, E5=50521, E6=2702765, E7=199360981
Famous constants by Infinite series:
Pi2/6 = 1/12 + 1/22 + 1/32 + 1/42 + ... [Euler, 1736] (Today this is known as Zeta[2])
Pi2/8 = 1/12 + 1/32 + 1/52 + 1/72 + ... [Euler, 1736]
Pi2/12 = 1/12 - 1/22 + 1/32 - 1/42 + ... [Euler, 1736]
2/Pi = (12*32*52*72*92*...)/(22*42*62*82*...) [Wallis, late 1600s]
4/Pi = 1+1/(2+32/(2+52/(2+72/(2+... continued fraction. [Wallis, late 1600s]
Zeta[4] = Pi4/60
Pi = 2/( sqrt(1/2) * sqrt(1/2 + (1/2)sqrt(1/2)) * sqrt(1/2+(1/2)sqrt(1/2+(1/2)sqrt(1/2))) * ... [François Viète, late 1500s] Significance: First infinite expression for calculating Pi.
Pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... [Leibniz, 1600s]
Pi/24 = sqrt(3)/32 + (1/12 - 1/160 - 1/3584 - 1/36864 - ...) [Sir Isaac Newton, late 1600s]
Misc. Theorems:
Rational Root Theorem: All rational solutions x=p/q of a polynomial equation anxn + an-1xn-1 + ... + a0, p is a factor of a0 and q is a factor of an
Misc. Facts and tidbits:
Ancient Pythagoreans first proved existence of irrational numbers, like sqrt(2).
Leonhard Euler (1707-1783) proved e and e2 were irrational in 1737.
Johann Heinrich Lambert (1728-1777) proved Pi was irrational in 1761.
Joseph Liouville (1809-1882) was first to find a transcendental number in 1844. In 1851 he discovered a special class of them which are called 'Liouville Numbers'.
Charles Hermite (1822-1901) proved 'e' is transcendental in 1873.
Ferdinand von Lindemann (1852-1939) proved that Pi is transcendental in 1882.
Gelfond proved ePi and 2sqrt(2) were transcendental and in 1934 proved the general case known as Gelfond's theorem.
It isn't known whether PiPi or ee is transcendental. Here lies a chance for fame!
No proof exists as of yet whether Pie or 2e or 2Pi are even irrational, much less transcendental.
sin(1 radian) is transcendental. sin(1 degree) is algebraic.
Leibniz was the first to publish on Differential Calculus, and is who we owe the d/dx syntax. However, Sir Isaac Newton was also aware of the technique, and there is a hot debate as to whether Newton or Leibniz is the father of the Differential Calculus.
Joe K. Crump |
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