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Mathematics Crib Sheet
Mathematical Formulae
(a/sin(A)) = (b/sin(B)) = (c/sin(C)) = 2r
sin(½A) = ((s-b)(s-c)/bc)½       s=½(a+b+c)
cos(½A) = (s(s-a)/bc)½       s=½(a+b+c)
tan(½A) = ((s-b)(s-c)/(s(s-a)))½       s=½(a+b+c)
tan(½(B-C)) = ((b-a)/(b+c))cot(½A)
sin(-A) = -sin(A)
cos(-A) = cos(A)
cosec(A) = 1/sin(A)
sec(A) = 1/cos(A)
cot(A) = 1/tan(A)
sin2(A) + cos2(A) = 1
cos2(A) = (1 + cos(2A))/2
sin2(A) = (1 - cos(2A))/2
sin(A)cos(A) = (sin(2A))/2 sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
sin(A-B) = sin(A)cos(B) - cos(A)sin(B)
cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
cos(A-B) = cos(A)cos(B) + sin(A)sin(B)
sec2(A) = 1 + tan2(A)
cosec2(A) = 1 + cot2(A)
sin(2A) = 2sin(A)cos(A)
cos(2A) = 2cos2(A) - 1
cos(2A) = 1 - 2sin2(A)
tan(2A) = (2tan(A))/(1-tan2(A))
tan(A+B) = (tan(A)+tan(B))/(1-tan(A)tan(B))
tan(A-B) = (tan(A)-tan(B))/(1+tan(A)tan(B))
sin(3A) = 3sin(A)-4sin3(A)
cos(3A) = 4cos3(A)-3cos(A)
tan(3A) = (3tan(A)-tan3(A))/(1-3tan(A))
sin(2A) = 2tan(A)/(1+tan2(A))
cos(2A) = (1-tan2(A))/(1+tan2(A))
sin(A)+sin(B) = 2sin(½(A+B))cos(½(A-B))
sin(A)-sin(B) = 2cos(½(A+B))sin(½(A-B))
cos(A)+cos(B) = 2cos(½(A+B))cos(½(A-B))
cos(A)-cos(B) = 2sin(½(A+B))sin(½(B-A))
tan(A)+tan(B) = sin(A+B)/(cos(A)cos(B))
tan(A)-tan(B) = sin(A-B)/(cos(A)cos(B))
cos(A+B)+cos(A-B) = 2cos(A)cos(B)
cos(A-B)-cos(A+B) = 2sin(A)sin(B)
sinh(x) = (ex-e-x) / 2
cosh(x) = (ex+e-x) / 2
tanh(x) = (ex-e-x) / (ex+e-x)
sech(x) = 2 / (ex-e-x)
cosh(x) = 2 / (ex+e-x)
tanh(x) = (ex-e-x) / (ex+e-x)
sin(x) = (x1/1!) - (x3/3!) + (x5/5!) - ...
cos(x) = (x0/0!) - (x2/2!) + (x4/4!) - ...
ex = (x0/0!) + (x1/1!) + (x2/2!) + ...
sinh(x) = (x1/1!) + (x3/3!) + (x5/5!) + ...
cosh(x) = (x0/0!) + (x2/2!) + (x4/4!) + ...
log(1+x) = (x1/1!) + (x2/2!) + (x3/3!) + ...
(x+1)n = 1 + (nx1/1!) + (n(n-1)x2/2!) + (n(n-1)(n-2)x3/3!) + ...
d/dxsin(x) = cos(x)
d/dxcos(x) = -sin(x)
d/dxtan(x) = sec2(x)
d/dxcot(x) = -cosec2(x)
d/dxsec(x) = sec(x)tan(x)
d/dxcosec(x) = -cosec(x)cot(x)
d/dxsinh(x) = cosh(x)
d/dxcosh(x) = sinh(x)
d/dxtanh(x) = sech2(x)
d/dxcosech(x) = -coth(x)cosech(x)
d/dxsech(x) = -tanh(x)sech(x)
d/dxcoth(x) = -cosech2(x)
d/dxaxn = anx(x-1)
d/dxeax = aeax
d/dxax = axloge(a)
d/dxxx = xx/(1+loge(x))
d/dxloge(x) = 1/x
d/dxloga(x) = 1/(x loga(e))
d/dxsin-1(x/a) = 1/(a2-x2)½
d/dxcos-1(x/a) = -1/(a2-x2)½
d/dxtan-1(x/a) = a/(a2+x2)
d/dxeax = aeax
xn = x(n+1)/(n+1) +c
1/(x2+n2) =(1/n) tan-1(x/n) +c
1/(x2-n2) =(1/2n)loge((x-n)/(x+n)) +c
1/(a+bx) = 1/(b loge(a+bx)) +c
x/(ax+b) = (ax + (1-loge(ax+b))b)/a2 +c
1/x =loge(x) +c
1/(x2-a2)½ = cosh-1(x/a) +c
1/(x2+a2)½ = sinh-1(x/a) +c
ex = ex +c
1/(a2-x2) = (1/a) tanh-1(x/a) +c
ax = ax/loge(a) +c
xax = (ax/loge(a))-(ax/(loge(a))2) +c
xeax = (eax(ax-1))/a2 +c
loge(x) = x(loge(x)-1) +c
1/xloge(x) =loge(loge(x) ) +c
sin(x) = -cos(x) +c
cos(x) = sin(x) +c
tan(x) = -loge(cos(x)) +c
cotan(x) =loge(sin(x)) +c
sec(x) =loge(sec(x) + tan(x)) +c
cosec(x) = loge(tan(x/2)) +c
sinh(x) = cosh(x) +c
cosh(x) = sinh(x) +c
tanh(x) = loge(cosh(x)) +c
cosech(x) =loge(tanh(x/s)) +c
sech(x) = tan-1(sinh(x)) +c
coth(x) =loge(sinh(x))
sinh2(x) = (-x/2) + (sinh(2x)/4) +c
cosh2(x) = (x/2) + (sinh(2x)/4) +c
sech2(x) = tanh(x) +c
cosech2(x) = -coth(x) +c
tanh2(x) = x - tanh(x) +c
anx(n-1) = axn +c
aeax = eax +c
a(cos(ax)) = sin(ax) +c
a(sin(ax)) = -cos(ax) +c
a(sec2(ax)) = tan(ax) +c
1/(1-x2)½ = sin-1(x) +c
1/(1+x2) = tan-1(x)
x2+y2+2ax+2by+c = 0     O=(-a, -b)     r=(a2+b2-c)½
(x2/a2)+(y2/b2) = 1
y = x2+mx+c
y = mx+c
x3+y3 = (x+y)(x2-xy+y2)
x3-y3 = (x-y)(x2+xy+y2)
x = (-b±(b2-4ac)½)/2a
loge(xy) = loge(x)+loge(y)
loge(x/y) = loge(x)-loge(y)
loge(xy) = y(loge(x))
(loge(x))(logx(e)) = 1
loge(x) = (loga(x))(loge(a))
i = (-1)½
a2+b2=c2
ax ay=ax+y
eix=cos(x)+i sin(x)
r(cos(x)+i sin(x))y=ry(cos(yx)+i sin(yx))
x=n(n+1)/2
x2=n(n+1)(2n+1)/6
x3=n2(n+1)2/4
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