MATH FORMULAS
k is an integer
± means "plus or minus"
∓ means "minus or plus"
≠ means "does not equal"
√( means square root of the argument
abs( means the absolute value of the argument
i means the √(-1), or the imaginary number
π means the number pi

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Trigonometric Functions

Definition	Ratio
sin(x)= (exi-e-xi)/(2i)	tan(x)= sin(x)/cos(x)
cos(x)= (exi+e-xi)/2	cot(x)= cos(x)/sin(x)

Reciprocal
sin(x)= 1/csc(x)	csc(x)= 1/sin(x)	sin(x)csc(x)=1
cos(x)= 1/sec(x)	sec(x)= 1/cos(x)	cos(x)sec(x)=1
tan(x)= 1/cot(x)	cot(x)= 1/tan(x)	tan(x)cot(x)=1

Pythagorean
sin²(x)+cos²(x)=1	csc²(x)-cot²(x)=1	sec²(x)-tan²(x)=1
sin²(x)= 1-cos²(x)	csc²(x)= cot²(x)+1	sec²(x)= tan²(x)+1
cos²(x)= 1-sin²(x)	cot²(x)= csc²(x)-1	tan²(x)= sec²(x)-1

Cofunction
sin(x)= cos(x-π/2) = cos(π/2-x)	cos(x)= sin(π/2±x) = cos(-x)	tan(x)= cot(π/2-x) = -cot(x±π/2)
csc(x)= sec(x-π/2) = sec(π/2-x)	sec(x)= csc(π/2±x) = sec(-x)	cot(x)= tan(π/2-x) = -tan(x±π/2)

Sum/Difference
sin(A±B)= sin(A)cos(B)±cos(A)sin(B)
cos(A±B)= cos(A)cos(B)∓sin(A)sin(B)
tan(A±B)= [tan(A)±tan(B)]/[1∓tan(A)tan(B)]

Product/Sum	Sum/Product
sin(A)cos(B)= [sin(A+B)+sin(A-B)]/2	sin(A)±sin(B)= 2sin[(A±B)/2]cos[(A∓B)/2]
cos(A)sin(B)= [sin(A+B)-sin(A-B)]/2
sin(A)sin(B)= [cos(A-B)-cos(A+B)]/2	cos(A)+cos(B)= 2cos[(A+B)/2]cos[(A-B)/2]
cos(A)cos(B)= [cos(A+B)+cos(A-B)]/2	cos(A)-cos(B)= -2sin[(A+B)/2]sin[(A-B)/2]

Double Angle
sin(2x)= 2sin(x)cos(x)
cos(2x)= cos²(x)-sin²(x) = 2cos²(x)-1 = 1-2sin²(x)
tan(2x)= 2tan(x)/[1-tan²(x)] = 2/[cot(x)-tan(x)] = 2cot(x)/[cot²(x)-1]	x ≠ (1+2k)π/4

Triple Angle
sin(3x)= 3sin(x)-4sin³(x)
cos(3x)= 4cos³(x)-3cos(x)
tan(3x)= [3tan(x)-tan³(x)]/[1-3tan²(x)]	x ≠ (1+2k)π/6

Power Reducing
sin²(x)= [1-cos(2x)]/2
cos²(x)= [1+cos(2x)]/2
tan²(x)= [1-cos(2x)]/[1+cos(2x)] = 1-2tan(x)/tan(2x)	x ≠ (1+2k)π/2

Half Angle
sin(x/2)= ±√([1-cos(x)]/2)	use (-) when x/2 is in Quadrant 3 or 4
cos(x/2)= ±√([1+cos(x)]/2)	use (-) when x/2 is in Quadrant 2 or 3
tan(x/2)= ±√([1-cos(x)]/[1+cos(x)])	use (-) when x/2 is in Quadrant 2 or 4	x ≠ (1+2k)π

The conditions for the Double Angle, Triple Angle, Half Angle, and Power Reducing formulas involving tangent only apply to the original term on the left.

Sine Law	Cosine Law
sin(A)/a = sin(B)/b = sin(C)/c	a²= b²+c²-(2bc)cos(A)
	b²= a²+c²-(2ac)cos(B)
	c²= a²+b²-(2ab)cos(C)

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Hyperbolic Functions

Definition	Ratio
sinh(x)= (ex-e-x)/2	tanh(x)= sinh(x)/cosh(x)
cosh(x)= (ex+e-x)/2	coth(x)= cosh(x)/sinh(x)

Reciprocal
sinh(x)= 1/csch(x)	csch(x)= 1/sinh(x)	sinh(x)csch(x)=1
cosh(x)= 1/sech(x)	sech(x)= 1/cosh(x)	cosh(x)sech(x)=1
tanh(x)= 1/coth(x)	coth(x)= 1/tanh(x)	tanh(x)coth(x)=1

Pythagorean
cosh²(x)-sinh²(x)=1	coth²(x)-csch²(x)=1	sech²(x)+tanh²(x)=1
cosh²(x)= sinh²(x)+1	coth²(x)= csch²(x)+1	sech²(x)= 1-tanh²(x)
sinh²(x)= cosh²(x)-1	csch²(x)= coth²(x)-1	tanh²(x)= 1-sech²(x)

Sum/Difference
sinh(A±B)= sinh(A)cosh(B)±cosh(A)sinh(B)
cosh(A±B)= cosh(A)cosh(B)±sinh(A)sinh(B)
tanh(A±B)= [tanh(A)±tanh(B)]/[1±tanh(A)tanh(B)]

Product/Sum	Sum/Product
sinh(A)cosh(B)= [sinh(A+B)+sinh(A-B)]/2	sinh(A)±sinh(B)= 2sinh[(A±B)/2]cosh[(A∓B)/2]
cosh(A)sinh(B)= [sinh(A+B)-sinh(A-B)]/2
sinh(A)sinh(B)= [cosh(A+B)-cosh(A-B)]/2	cosh(A)+cosh(B)= 2cosh[(A+B)/2]cosh[(A-B)/2]
cosh(A)cosh(B)= [cosh(A+B)+cosh(A-B)]/2	cosh(A)-cosh(B)= 2sinh[(A+B)/2]sinh[(A-B)/2]

Double Angle
sinh(2x)= 2sinh(x)cosh(x)
cosh(2x)= cosh²(x)+sinh²(x) = 2sinh²(x)+1 = 2cosh²(x)-1
tanh(2x)= 2tanh(x)/[1+tanh²(x)] = 2/[tanh(x)+coth(x)] = 2coth(x)/[coth²(x)+1]

Triple Angle
sinh(3x)= sinh³(x)+3cosh²(x)sinh(x)
cosh(3x)= cosh³(x)+3sinh²(x)cosh(x)
tanh(3x)= [3tanh(x)+tanh³(x)]/[1+3tanh²(x)]

Power Reducing
sinh²(x)= [cosh(2x)-1]/2
cosh²(x)= [cosh(2x)+1]/2
tanh²(x)= [cosh(2x)-1]/[cosh(2x)+1] = 2tanh(x)/tanh(2x)-1

Inverse Hyperbolic Functions
arcsinh(x)= ln[x+√(x²+1)]	x is a real number
arccosh(x)= ln[x±√(x²-1)]	x > 1
arctanh(x)= ln[(1+x)/(1-x)]/2	-1 < x < 1
arccsch(x)= ln[(1±√(x²+1))/x]	x ≠ 0; use (+) when x > 0; use (-) when x < 0
arcsech(x)= ln[(1±√(1-x²))/x]	0 < x < 1
arccoth(x)= ln[(x+1)/(x-1)]/2	abs(x) > 1

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Calculus
u = f(x)
v = g(x)
u′ = du/dx = f′(x)
v′ = dv/dx = g′(x)
_a^b∫( means the integral of the argument with lower and upper bounds a and b (if any)
log_a( means the logarithm of base a
ln( means the natural logarithm of base e
Derivatives

Derivatives
Power Rule	Product Rule	Quotient Rule	Exponential, Logarithmic
un	u v	u/v	eu	au	ln(u)	loga(u)
nun-1u′	uv′+vu′	(u′v-v′u)/v²	euu′	auln(a)u′	u′/u	u′/[u ln(a)]

Derivatives of Trigonometric Functions
sin(u)	cos(u)	tan(u)	csc(u)	sec(u)	cot(u)
cos(u)u′	-sin(u)u′	sec²(u)u′	-csc(u)cot(u)u′	sec(u)tan(u)u′	-csc²(u)u′

Derivatives of Inverse Trigonometric Functions
arcsin(u)	arccos(u)	arctan(u)	arccsc(u)	arcsec(u)	arccot(u)
u′/√(1-u²)	-u′/√(1-u²)	u′/(1+u²)	-u′/[u √(u²-1)]	u′/[u √(u²-1)]	-u′/(1+u²)

Derivatives of Hyperbolic Functions
sinh(u)	cosh(u)	tanh(u)	csch(u)	sech(u)	coth(u)
cosh(u)u′	sinh(u)u′	sech²(u)u′	-csch(u)coth(u)u′	-sech(u)tanh(u)u′	-csch²(u)u′

Derivatives of Inverse Hyperbolic Functions
arcsinh(u)	arccosh(u)	arctanh(u)	arccsch(u)	arcsech(u)	arccoth(u)
u′/√(1+u²)	u′/√(u²-1)	u′/(1-u²)	-u′/[abs(u)√(u²+1)]	-u′/[u √(1-u²)]	u′/(1-u²)

Integrals
note: an arbitrary constant is to be added to each formula

Basic Forms
undu	u dv	eudu	du/u	audu
un+1/(n+1), n ≠ -1	uv - ∫(v du)	eu	ln(u)	au/ln(a), a > 0, a ≠ 1

Basic Trigonometric Forms
tan(u)du	cot(u)du	sec(u)du	csc(u)du
ln[abs(sec(u))]	ln[abs(sin(u))]	ln[abs(sec(u)+tan(u))]	ln[abs(csc(u)-cot(u))]

Basic Hyperbolic Forms
tanh(u)du	coth(u)du	sech(u)du	csch(u)du
ln[cosh(u)]	ln[abs(sinh(u))]	arctan[abs(sinh(u))]	ln[abs(tanh(u/2))]

More Basic Forms
du/(u²+a²)	du/(u²-a²), u² ≠ a²	du/(a²-u²), u² ≠ a²
arctan(u/a)/a, a ≠ 0	ln[abs((u-a)/(u+a))]/(2a), a ≠ 0	ln[abs((a+u)/(a-u))]/(2a), a ≠ 0

More Basic Forms
du/√(a²-u²), a² > u²	du/[u √(u²-a²)], u² > a² > 0
arcsin(u/a)	arcsec(a/u)/a

du/[(a+bu)(m+nu)]	(u du)/[(a+bu)(m+nu)]
ln[(m+nu)/(a+bu)]/(an-bm), an ≠ bm	[ln[a+bu](a/b)-ln[m+nu](m/n)]/(an-bm), an ≠ bm

Exponential Forms
u eaudu	uneaudu	eausin(bu)du	eaucos(bu)du
[(au-1)eau]/a²	[uneau]/a - (n/a)∫(un-1eaudu)	eau[a sin(bu) - b cos(bu)]/(a²+b²)	eau[a cos(bu) - b sin(bu)]/(a²+b²)

Logarithmic Forms
ln(u)du	du/[u ln(u)]	unln(u)du
u ln(u) - u	ln[abs(ln(u))]	un+1[(n+1)ln(u)-1]/(n+1)², n ≠ -1

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Laplace Transforms
∞ = infinity
* = the symbol for convolucence

Properties
L{f(t)}(s) = 0∞∫( e-stf(t)dt ) = F(s)
L-1{F(s)}(t) = f(t)
L{eatf(t)}(s) = F(s-a)
L{f′(t)}(s) = sF(s)-f(0)
L{f″(t)}(s) = s²F(s)-sf(0)-f′(0)
L{f(n)(t)}(s) = snF(s)-sn-1f(0)-sn-2f′(0)-...-sf(n-2)(0)-f(n-1)(0)
L{tnf(t)}(s) = (-1)nF(n)(s)
f(t)*g(t) = 0t∫( f(t-v)g(v)dv )
L{f(t)*g(t)}(s) = F(s)G(s)
L-1{F(s)G(s)}(t) = f(t)*g(t)

Brief Table
f(t)	F(s)
1	1/s; s > 0
eat	1/(s-a); s > a
tn; n=1,2,3,...	n!/(sn+1); s > 0
eattn; n=1,2,3,...	n!/[(s-a)n+1]; s > a
sin(bt)	b/(s²+b²); s > 0
cos(bt)	s/(s²+b²); s > 0
eatsin(bt)	b/[(s-a)²+b²]; s > a
eatcos(bt)	(s-a)/[(s-a)²+b²]; s > a
t sin(bt)	2bs/(s²+b²)²
t cos(bt)	(s²-b²)/(s²+b²)²
sinh(bt)	b/(s²-b²)
cosh(bt)	s/(s²-b²)

sin(bt)-(bt)cos(bt)	2b³/(s²+b²)²
sin(bt)+(bt)cos(bt)	2bs²/(s²+b²)²
sin(bt)cosh(bt)-cos(bt)sinh(bt)	4b³/(s4+4b4)
sin(bt)sinh(bt)	2b²s/(s4+4b4)
sinh(bt)-sin(bt)	2b³/(s4-b4)
cosh(bt)-cos(bt)	2b²s/(s4-b4)

eat-ebt	(a-b)/[(s-a)(s-b)]
aeat-bebt	s(a-b)/[(s-a)(s-b)]
√(1/t)	√[π/s]
√(t)	√[π/s]/(2s)
tn-½; n=1,2,3,...	[(1)(3)(5)...(2n-1)√π]/[2nsn+½]
tr; r > -1	Γ(r+1)/[sr+1]

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