Trigonometric Properties and Identities Cheat Sheet

Circular Functions Definitions

Name	Right-Triangle Definition	Domain	Range
Sine Function	sin(θ)=o/h	(-∞,∞)	[-1,1]
Cosine Function	cos(θ)=a/h	(-∞,∞)	[-1,1]
Tangent Function	tan(θ)=o/a	{θ\|θ≠π/2±πn}	(-∞,∞)
Cosecant Function	csc(θ)=h/o	{θ\|θ≠±πn}	(-∞,-1]∪[1,∞)
Secant Function	sec(θ)=h/a	{θ\|θ≠π/2±πn}	(-∞,-1]∪[1,∞)
Cotangent Function	cot(θ)=a/o	{θ\|θ≠±πn}	(-∞,∞)
Inverse Sine Function	arcsin(o/h)=θ	[-1,1]	[-π/2,π/2]
Inverse Cosine Function	arccos(a/h)=θ	[-1,1]	[0,π]
Inverse Tangent Function	arctan(o/a)=θ	(-∞,∞)	(-1,1)
Inverse Cosecant Function	arccsc(h/o)=θ	(-∞,-1)∪(1,∞)	[-π/2,0)∪(0,π/2]
Inverse Secant Function	arcsec(h/a)=θ	(-∞,-1)∪(1,∞)	[0,π/2)∪(π/2,π]
Inverse Cotangent Function	arccot(a/o)=θ	(-∞,∞)	(0,1)
Circular Euler Relation	e^±iθ=cos(θ)±isin(θ)
De Moivre's Theorem	e^inθ=(cos(θ)+isin(θ))ⁿ=cos(nθ)+isin(nθ)

n ∈ ℕ₁ = {1,2,3,4,5,...}
"h" is the "hypotenuse" leg of a right triangle. It is directly across the right (90°) angle, and it has the longest length of the three sides.
"o" is the "opposite" leg of a right triangle. It is directly across the chosen angle θ.
"a" is the "adjacent" leg of a right triangle. It is the leg that is neither the hypotenuse leg, nor the opposite leg.
By the Pythagorean theorem, o²+a²=h²

Hyperbolic Functions Definitions

Name	Exponential Definition	Domain	Range
Hyperbolic Sine Function	sinh(θ)=(e^θ-e^-θ)/2	(-∞,∞)	(-∞,∞)
Hyperbolic Cosine Function	cosh(θ)=(e^θ+e^-θ)/2	(-∞,∞)	[1,∞)
Hyperbolic Tangent Function	tanh(θ)=(e^θ-e^-θ)/(e^θ+e^-θ)	(-∞,∞)	(-1,1)
Hyperbolic Cosecant Function	csch(θ)=2/(e^θ-e^-θ)	(-∞,0)∪(0,∞)	(-∞,0)∪(0,∞)
Hyperbolic Secant Function	sech(θ)=2/(e^θ+e^-θ)	(-∞,∞)	(0,1]
Hyperbolic Cotangent Function	coth(θ)=(e^θ+e^-θ)/(e^θ-e^-θ)	(-∞,0)∪(0,∞)	(-∞,-1)∪(1,∞)
Inverse Hyperbolic Sine Function	arcsinh(x)=ln(x+√(x²+1))	(-∞,∞)	(-∞,∞)
Inverse Hyperbolic Cosine Function	arccosh(x)=ln(x+√(x²-1))	[1,∞)	[0,∞)
Inverse Hyperbolic Tangent Function	arctanh(x)=½ln((1+x)/(1-x))	(-1,1)	(-∞,∞)
Inverse Hyperbolic Cosecant Function	arccsch(x)=ln((1±√(1+x²))/x)	(-∞,0)∪(0,∞)	(-∞,0)∪(0,∞)
Inverse Hyperbolic Secant Function	arcsech(x)=ln((1+√(1-x²))/θ)	(0,1]	[0,∞)
Inverse Hyperbolic Cotangent Function	arccoth(x)=½ln((x+1)/(x-1))	(-∞,-1)∪(1,∞)	(-∞,0)∪(0,∞)
Hyperbolic Euler Relation	e^±θ=cosh(θ)±sinh(θ)
De Moivre's Theorem (Hyperbolic)	e^nθ=(cosh(θ)+sinh(θ))ⁿ=cosh(nθ)+sinh(nθ))

n ∈ ℕ₁ = {1,2,3,4,5,...}

Complex Definitions

Name	Complex Relation	Circular-Hyperbolic Relation
Complex Sine	sin(z)=(e^iz-e^-iz)/2i	sin(z)=-isinh(iz)
Complex Cosine	cos(z)=(e^iz+e^-iz)/2	cos(z)=cosh(iz)
Complex Tangent	tan(z)=-i(e^iz-e^-iz)/(e^iz+e^-iz)	tan(z)=-itanh(iz)
Complex Cosecant	csc(z)=2i/(e^iz-e^-iz)	csc(z)=icsch(iz)
Complex Secant	sec(z)=2/(e^iz+e^-iz)	sec(z)=sech(iz)
Complex Cotangent	cot(z)=i(e^iz+e^-iz)/(e^iz-e^-iz)	cot(z)=icoth(iz)
Complex Inverse Sine	arcsin(z)=-iln(iz±√(1-z²))	arcsin(z)=-iarcsinh(iz)
Complex Inverse Cosine	arccos(z)=-iln(z±i√(1-z²))	arccos(z)=±iarccosh(z)
Complex Inverse Tangent	arctan(z)=(i/2)ln((i+z)/(i-z))	arctan(z)=-iarctanh(iz)
Complex Inverse Cosecant	arccsc(z)=-iln((i+√(z²-1))/z)	arccsc(z)=iarccsch(iz)
Complex Inverse Secant	arcsec(z)=-iln((1+√(1-z²))/z)	arcsec(z)=±iarcsech(z)
Complex Inverse Cotangent	arccot(z)=-(i/2)ln((z+i)/(z-i))	arccot(z)=±iarccoth(iz)
Complex Hyperbolic Sine	None	sinh(z)=-isin(iz)
Complex Hyperbolic Cosine	None	cosh(z)=cos(iz)
Complex Hyperbolic Tangent	None	tanh(z)=-itan(iz)
Complex Hyperbolic Cosecant	None	csch(z)=icsc(iz)
Complex Hyperbolic Secant	None	sech(z)=sec(iz)
Complex Hyperbolic Cotangent	None	coth(z)=icot(iz)
Complex Inverse Hyperbolic Sine	None	arcsinh(z)=-iarcsin(iz)
Complex Inverse Hyperbolic Cosine	None	arccosh(z)=±iarccos(z)
Complex Inverse Hyperbolic Tangent	None	arctanh(z)=-iarctan(iz)
Complex Inverse Hyperbolic Cosecant	None	arccsch(z)=-iarccsc(iz)
Complex Inverse Hyperbolic Secant	None	arcsech(z)=±iarcsec(z)
Complex Inverse Hyperbolic Cotangent	None	arccoth(z)=-iarccot(iz)

i=√(-1)
z is a complex variable of the form a+bi, where a and b are real numbers, and i is the imaginary unit

Circular Functions Unit Circle Values

θ (Radians)	θ (Degrees)	*sin(θ)*	*cos(θ)*	*tan(θ)*	csc(θ)	sec(θ)	cot(θ)
0	0°	0	1	0	undefined	1	undefined
π/6	30°	*1/2*	√3/2	√3/3	2	2√3/3	√3
π/4	45°	√2/2	√2/2	1	√2	√2	1
π/3	60°	√3/2	*1/2*	√3	2√3/3	2	√3/3
π/2	90°	1	0	*undefined*	1	undefined	0

2π/3	120°	√3/2	-1/2	-√3	2√3/3	-2	-√3/3
3π/4	135°	√2/2	-√2/2	-1	√2	-√2	-1
5π/6	150°	1/2	-√3/2	-√3/3	2	-2√3/3	-√3
π	180°	0	-1	0	undefined	-1	undefined

7π/6	210°	-1/2	-√3/2	√3/3	-2	-2√3/3	√3
5π/4	225°	-√2/2	-√2/2	1	-√2	-√2	1
4π/3	240°	-√3/2	-1/2	√3	-2√3/3	-2	√3/3
3π/2	270°	-1	0	undefined	-1	undefined	0

5π/3	300°	-√3/2	1/2	-√3	-2√3/3	2	-√3/3
7π/4	315°	-√2/2	√2/2	-1	-√2	√2	-1
11π/6	330°	-1/2	√3/2	-√3/3	-2	2√3/3	-√3
2π	360°	0	1	0	undefined	1	undefined

The coordinates (cos(θ), sin(θ)) represent x and y coordinates of θ on the unit circle x²+y²=1

Circular Compositional Identities

Composition	sin(x)	cos(x)	tan(x)
arcsin(x)	x	√(1-x²)	x/√(1-x²)
arccos(x)	√(1-x²)	x	√(1-x²)/x
arctan(x)	x/√(1+x²)	1/√(1+x²)	x
arccsc(x)	1/x	√(x²-1)/\|x\|	±1/√(x²-1)
arcsec(x)	√(x²-1)/\|x\|	1/x	±√(x²-1)
arccot(x)	1/√(1+x²)	x/√(1+x²)	1/x

Each composition is valid on different domains

Hyperbolic Compositional Identities

Composition	sinh(x)	cosh(x)	tanh(x)
arcsinh(x)	x	√(1+x²)	x/√(1-x²)
arccosh(x)	√(x²-1)	x	√(x²-1)/x
arctanh(x)	x/√(1-x²)	1/√(1-x²)	x
arccsch(x)	1/x	√(x²+1)/\|x\|	1/√(x²+1)
arcsech(x)	√(1-x²)/x	1/x	√(1-x²)
arccoth(x)	x/√(1-x²)	\|x\|/√(x²-1)	1/x

Each composition is valid on different domains

Circular Quotient & Reciprocal Identities

Tangent Quotient	tan(θ)=sin(θ)/cos(θ)
Cotangent Quotient	cot(θ)=cos(θ)/sin(θ)

Sine Reciprocal	sin(θ)=1/csc(θ)
Cosine Reciprocal	cos(θ)=1/sec(θ)
Tangent Reciprocal	tan(θ)=1/cot(θ)
Cosecant Reciprocal	csc(θ)=1/sin(θ)
Secant Reciprocal	sec(θ)=1/cos(θ)
Cotangent Reciprocal	cot(θ)=1/tan(θ)

All the following identities are true for values that do not cause division by zero

Cofunctional Phase Shift Properties

Sine Complimentary	sin(θ)=cos(π/2-θ)
Sine Supplementary	sin(θ)=sin(π-θ)
Cosine Complimentary	cos(θ)=sin(π/2-θ)
Cosine Supplementary	cos(θ)=-cos(π-θ)
Tangent Complimentary	tan(θ)=cot(π/2-θ)
Tangent Supplementary	tan(θ)=-tan(πn-θ)
Cosecant Complimentary	csc(θ)=sec(π/2-θ)
Cosecant Supplementary	csc(θ)=csc(π-θ)
Secant Complimentary	sec(θ)=csc(π/2-θ)
Secant Supplementary	sec(θ)=-sec(π-θ)
Cotangent Complimentary	cot(θ)=tan(π/2-θ)
Cotangent Supplementary	cot(θ)=-cot(πn-θ)

n ∈ ℕ₁ = {1,2,3,4,5,...}

Periodicity Properties

Sine Periodicity	sin(θ)=sin(θ±2πn)
Cosine Periodicity	cos(θ)=cos(θ±2πn)
Tangent Periodicity	tan(θ)=tan(θ±πn)
Cosecant Periodicity	csc(θ)=csc(θ±2πn)
Secant Periodicity	sec(θ)=sec(θ±2πn)
Cotangent Periodicity	cot(θ)=cot(θ±πn)

n ∈ ℕ₁ = {1,2,3,4,5,...}

Circular Parity Properties

Sine Odd	sin(-θ)=-sin(θ)
Cosine Even	cos(-θ)=cos(θ)
Tangent Odd	tan(-θ)=-tan(θ)
Cosecant Odd	csc(-θ)=-csc(θ)
Secant Even	sec(-θ)=sec(θ)
Cotangent Odd	cot(-θ)=-cot(θ)

Circular Pythagorean Identities

Sine-Cosine Pythagorean	sin²(θ)+cos²(θ)=1
Secant-Tangent Pythagorean	tan²(θ)+1=sec²(θ)
Cosecant-Cotangent Pythagorean	1+cot²(θ)=csc²(θ)

The last two Pythagorean Identities are obtained by dividing all the terms of the original Sine-Cosine Identity by cos²(θ) and sin²(θ), respectively

(C) Half/Multiple-Angle Identities

Sine Half-Angle	sin(θ/2)=±√(½(1-cos(θ)))
Cosine Half-Angle	cos(θ/2)=±√(½(1+cos(θ)))
Tangent Half-Angle 1	tan(θ/2)=±√((1-cos(θ))/(1+cos(θ)))
Tangent Half-Angle 2	tan(θ/2)=(1-cos(θ))/sin(θ)
Tangent Half-Angle 3	tan(θ/2)=sin(θ)/(1+cos(θ))

Sine Double-Angle 1	sin(2θ)=2sin(θ)cos(θ)
Sine Double-Angle 2	sin(2θ)=2tan(θ)/(1+tan²(θ))
Cosine Double-Angle 1	cos(2θ)=cos²(θ)-sin²(θ)
Cosine Double-Angle 2	cos(2θ)=2cos²(θ)-1
Cosine Double-Angle 3	cos(2θ)=1-2sin²(θ)
Cosine Double-Angle 4	cos(2θ)=(1-tan²(θ))/(1+tan²(θ))
Tangent Double-Angle 1	tan(2θ)=2tan(θ)/(1-tan²(θ))
Tangent Double-Angle 2	tan(2θ)=2/(cot(θ)-tan(θ))

Sine Triple-Angle	sin(3θ)=3sin(θ)-4sin³(θ)
Cosine Triple-Angle	cos(3θ)=4cos³(θ)-3cos(θ)
Tangent Triple-Angle	tan(3θ)=(3tan(θ)-tan³(θ))/(1-3tan²(θ))

Sine Multiple-Angle Formula: sin(nθ)=∑ⁿ_k=0 (ⁿ_k)cos^k(θ)sin^n-k(θ)sin((π/2)(n-k))
Cosine Multiple-Angle Formula: cos(nθ)=∑ⁿ_k=0 (ⁿ_k)cos^k(θ)sin^n-k(θ)cos((π/2)(n-k))
All the following identities are true for values that do not cause division by zero

Circular Sum/Difference/Product Identities

Sine Sum/Difference	sin(θ±φ)	sin(θ)cos(φ)±cos(θ)sin(φ)
Sine Sum-Product	sin(θ)±sin(φ)	2sin((θ±φ)/2)cos((θ∓φ)/2)
Sine Product-Sum	sin(θ)sin(φ)	½(cos(θ-φ)-cos(θ+φ))
Cosine Sum/Difference	cos(θ±φ)	cos(θ)cos(φ)∓sin(θ)sin(φ)
Cosine Sum-Product	cos(θ)±cos(φ)	2cos((θ±φ)/2)cos((θ∓φ)/2)
Cosine Product-Sum	cos(θ)cos(φ)	½(cos(θ-φ)+cos(θ+φ))
Sine-Cosine Product-Sum	sin(θ)cos(φ)	½(sin(θ-φ)+sin(θ+φ))
Tangent Sum/Difference	tan(θ±φ)	(tan(θ)±tan(φ))/(1∓tan(θ)tan(φ))
Tangent Sum	tan(θ)±tan(φ)	sin(θ±φ)/(cos(θ)cos(φ))
Tangent Product	tan(θ)tan(φ)	(tan(θ)+tan(φ))/(cot(θ)+cot(φ))
Tangent-Cotangent Product	tan(θ)cot(φ)	(tan(θ)+cot(φ))/(cot(θ)+tan(φ))

Sine and Cosine Unit Circle

x²+y²=1

Circular-Inverse Reciprocal Identities

Sine Reciprocal	arcsin(1/x)=arccsc(x)
Cosine Reciprocal	arccos(1/x)=arcsec(x)
Tangent Reciprocal 1	arctan(1/x)=arccot(x), x>0
Tangent Reciprocal 2	arctan(1/x)=arccot(x)-π, x<0
Cosecant Reciprocal	arccsc(1/x)=arcsin(x)
Secant Reciprocal	arcsec(1/x)=arccos(x)
Cotangent Reciprocal 1	arccot(1/x)=arctan(x), x>0
Cotangent Reciprocal 2	arccot(1/x)=arctan(x)+π, x<0

Circular-Inverse Complimentary Identities

Sine Complimentary	arcsin(x)=π/2-arccos(x)
Cosine Complimentary	arccos(x)=π/2-arcsin(x)
Tangent Complimentary	arctan(x)=π/2-arccot(x)
Cosecant Complimentary	arccsc(x)=π/2-arcsec(x)
Secant Complimentary	arcsec(x)=π/2-arccsc(x)
Cotangent Complimentary	arccot(x)=π/2-arctan(x)

Circular-Inverse Negative Input Identities

Sine Odd	arcsin(-x)=-arcsin(x)
Cosine Translation	arccos(-x)=π-arccos(x)
Tangent Odd	arctan(-x)=-arctan(x)
Cosecant Odd	arccsc(-x)=-arccsc(x)
Secant Translation	arcsec(-x)=π-arcsec(x)
Cotangent Translation	arccot(-x)=π-arccot(x)

(CI) Half/Multiple Substitution Identities

Half Sine Substitution 1	½arcsin(x)=arcsin(√(1+x)/2))-π/4
Half Sine Substitution 2	½arcsin(x)=π/4-arcsin(√(1-x)/2))
Half Cosine Substitution 1	½arccos(x)=arccos(√((1+x)/2))
Half Cosine Substitution 2	½arccos(x)=π/2-arccos(√(1-x)/2))

Double Sine Substitution	2arcsin(x)=arcsin(2x√(1-x²)), \|x\|≤π/2
Double Cosine Substitution 1	2arccos(x)=arccos(2x²-1), x≥0
Double Cosine Substitution 2	2arccos(x)=2π-arccos(2x²-1), x≤0
Double Tangent Substitution 1	2arctan(x)=arcsin(2x/(1+x²)), \|x\|≤1
Double Tangent Substitution 2	2arctan(x)=±arccos((1-x²)/(1+x²))
Double Tangent Substitution 3	2arctan(x)=arctan(2x/(1-x²)), \|x\|<1

Triple Sine Substitution 1	3arcsin(x)=arcsin(3x-4x³), \|x\|≤½
Triple Sine Substitution 2	3arcsin(x)=arcsin(4x³-3x)±π, \|x\|≥½
Triple Cosine Substitution 1	3arccos(x)=arccos(3x-4x³), \|x\|≤½
Triple Cosine Substitution 2	3arccos(x)=arccos(4x³-3x)+π±π, \|x\|≥½
Triple Tangent Substitution 1	3arctan(x)=arctan((3x-x³)/(1-3x²)), \|x\|≤√3/3
Triple Tangent Substitution 2	3arctan(x)=arctan((3x-x³)/(1-3x²))±π, \|x\|≥√3/3

Circular-Inverse Sum/Difference Identities

Sine Sum/Difference	arcsin(x)±arcsin(y)=arcsin(x√(1-y²)±y√(1-x²)
Cosine Sum/Difference	arccos(x)±arccos(y)=arccos(xy∓√(1-x²)√(1-y²)
Cosine-Sine Sum/Difference	arccos(x)±arcsin(y)=arccos(x√(1-y²)∓y√(1-x²))
Tangent Sum/Difference	arctan(x)±arctan(y)=arctan((x±y)/(1∓xy)), 1∓xy≠0

Law of Sines/Cosines/Tangents

Law of Sines 1	sin(α)/a=sin(β)/b=sin(γ)/c
Law of Sines 2	a/sin(α)=b/sin(β)=c/sin(γ)
Law of Cosines 1	a²=b²+c²-2bccos(α)
Law of Cosines 2	b²=a²+c²-2accos(β)
Law of Cosines 3	c²=a²+b²-2abcos(γ)
Law of Tangents 1	(a-b)/(a+b)=tan((α-β)/2)/tan((α+β)/2)
Law of Tangents 2	(b-c)/(b+c)=tan((β-γ)/2)/tan((β+γ)/2)
Law of Tangents 3	(c-a)/(c+a)=tan((γ-α)/2)/tan((γ+α)/2)

Side lengths a, b, and c are opposite of the angles α, β, and γ, respectively.

Measurements And Formulas

Radians-Degrees	1 radian=180/π degrees; 1=(180/π)°
Degrees-Radians	1 degree=π/180 radians; 1°=π/180 radians
Degrees, Minutes, and Seconds (DMS)	1 degree=60 minutes=3600 seconds;1°=60'=3600''
Arc Length/Angular Displacement	s=rθ units
Sector Area	½r²θ units²
Area of a Triangle	A_T=½bh units²
Area of a Circle	A_C=πr² units²
Pythagorean Theorem	a²+b²=c²

Radians are unitless
a, b, and c are side lengths of a right-triangle

Tangent Unit Circle

(H) Quotient & Reciprocal Identities

Tangent Quotient	tanh(θ)=sinh(θ)/cosh(θ)
Cotangent Quotient	coth(θ)=cosh(θ)/sinh(θ)

Sine Reciprocal	sinh(θ)=1/csch(θ)
Cosine Reciprocal	cosh(θ)=1/sech(θ)
Tangent Reciprocal	tanh(θ)=1/coth(θ)
Cosecant Reciprocal	csch(θ)=1/sinh(θ)
Secant Reciprocal	sech(θ)=1/cosh(θ)
Cotangent Reciprocal	coth(θ)=1/tanh(θ)

All the following identities are true for values that do not cause division by zero

Hyperbolic Parity Properties

Sine Odd	sinh(-θ)=-sinh(θ)
Cosine Even	cosh(-θ)=cosh(θ)
Tangent Odd	tanh(-θ)=-tanh(θ)
Cosecant Odd	csch(-θ)=-csch(θ)
Secant Even	sech(-θ)=sech(θ)
Cotangent Odd	coth(-θ)=-coth(θ)

Hyperbolic Pythagorean Identities

Sine-Cosine Pythagorean	cosh²(θ)-sinh²(θ)=1
Secant Pythagorean	1-tanh²(θ)=sech²(θ)
Cosecant Pythagorean	coth²(θ)-1=csch²(θ)

The last two Hyperbolic Pythagorean Identities are obtained by dividing all the terms of the original Sine-Cosine Identity by cosh²(θ) and sinh²(θ), respectively

(H) Half-Angle & Multiple-Angle Identities

Sine Half-Angle	sinh(θ/2)=±√(½(cosh(θ)-1))
Cosine Half-Angle	cosh(θ/2)=√(½(cosh(θ)+1))
Tangent Half-Angle 1	tanh(θ/2)=±√((cosh(θ)-1)/(cosh(θ)+1))
Tangent Half-Angle 2	tanh(θ/2)=(cosh(θ)-1)/sinh(θ)
Tangent Half-Angle 3	tanh(θ/2)=sinh(θ)/(cosh(θ)+1)

Sine Double-Angle 1	sinh(2θ)=2sinh(θ)cosh(θ)
Sine Double-Angle 2	sinh(2θ)=2tanh(θ)/(1-tanh²(θ))
Cosine Double-Angle 1	cosh(2θ)=cosh²(θ)+sinh²(θ)
Cosine Double-Angle 2	cosh(2θ)=2cosh²(θ)-1
Cosine Double-Angle 3	cosh(2θ)=1+2sinh²(θ)
Cosine Double-Angle 4	cosh(2θ)=(1+tanh²(θ))/(1-tanh²(θ))
Tangent Double Angle 1	tanh(2θ)=2tanh(θ)/(1+tanh²(θ))
Tangent Double Angle 2	tanh(2θ)=2/(coth(θ)+tanh(θ))

Sine Triple-Angle	sinh(3θ)=3sinh(θ)+4sinh³(θ)
Cosine Triple-Angle	cosh(3θ)=4cosh³(θ)-3cosh(θ)
Tangent Triple-Angle	tanh(3θ)=(3tan(θ)+tan³(θ))/(1+3tan²(θ))

(H) Sum/Difference/Product Identities

Sine Sum/Difference	sinh(θ±φ)	sinh(θ)cosh(φ)±cosh(θ)sinh(φ)
Sine Sum-Product	sinh(θ)±sinh(φ)	2sinh((θ±φ)/2)cosh((θ∓φ)/2)
Sine Product-Sum	sinh(θ)sinh(φ)	½(cosh(θ+φ)-cosh(θ-φ))
Cosine Sum/Difference	cosh(θ±φ)	cosh(θ)cosh(φ)∓sinh(θ)sinh(φ)
Cosine Sum-Product 1	cosh(θ)+cosh(φ)	2cosh((θ+φ)/2)cosh((θ-φ)/2)
Cosine Sum-Product 2	cosh(θ)-cosh(φ)	2sinh((θ+φ)/2)sinh((θ-φ)/2)
Cosine Product-Sum	cosh(θ)cosh(φ)	½(cosh(θ+φ)+cosh(θ-φ))
Sine-Cosine Product-Sum	sinh(θ)cosh(φ)	½(sinh(θ+φ)+sinh(θ-φ))
Tangent Sum/Difference	tanh(θ±φ)	(tanh(θ)±tanh(φ))/(1±tanh(θ)tanh(φ))
Tangent Sum	tanh(θ)±tanh(φ)	sinh(θ±φ)/(cosh(θ)cosh(φ))
Tangent Product	tanh(θ)tanh(φ)	(tanh(θ)+tanh(φ))/(coth(θ)+coth(φ))
Tangent-Cotangent Product	tanh(θ)coth(φ)	(tanh(θ)+coth(φ))/(coth(θ)+tanh(φ))

Right-Triangle Relations

Hyperbolic-Inverse Reciprocal Identities

Sine Reciprocal	arcsinh(1/x)=arccsch(x)
Cosine Reciprocal	arccosh(1/x)=arcsech(x)
Tangent Reciprocal	arctanh(1/x)=arccoth(x)
Cosecant Reciprocal	arccsch(1/x)=arcsinh(x)
Secant Reciprocal	arcsech(1/x)=arccosh(x)
Cotangent Reciprocal	arccoth(1/x)=arctanh(x)

(HI) Negative Input Identities

Inverse Sine Odd	arcsinh(-x)=-arcsinh(x)
Inverse Tangent Odd	arctanh(-x)=-arctanh(x)
Inverse Cosecant Odd	arccsch(-x)=-arccsch(x)
Inverse Cotangent Odd	arccoth(-x)=-arccoth(x)

(HI) Half/Multiple Substitution Identities

Half Sine Substitution	½arcsinh(x)=±arcsinh(√(√((1+x²)-1)/2))
Half Cosine Substitution	½arccosh(x)=arccosh(√((x+1)/2))
Half Tangent Substitution	½arctanh(x)=arctanh(x/(1+√(1-x²)))

Double Sine Substitution 1	2arcsinh(x)=arcsinh(2x√(1+x²))
Double Sine Substitution 2	2arcsinh(x)=±arccosh(2x²+1)
Double Cosine Substitution	2arccosh(x)=arccosh(2x²-1), x≥1
Double Tangent Substitution	2arctanh(x)=arctanh(2x/(1+x²)), \|x\|<1

Triple Sine Substitution	3arcsinh(x)=arcsinh(3x+4x³)
Triple Cosine Substitution	3arccosh(x)=arccosh(4x³-3x)
Triple Tangent Substitution	3arctanh(x)=arctanh((3x+x³)/(1+3x²))

(HI) Sum/Difference Identities

Sine Sum/Difference	arcsinh(x)±arcsinh(y)=arcsinh(x√(y²+1)±y√(x²+1))
Cosine Sum/Difference	arccosh(x)±arccosh(y)=arccosh(xy±√((x²-1)(y²-1)))
Sine-Cosine Sum/Difference 1	arcsinh(x)±arccosh(y)=arcsinh(xy±√((x²+1)(y²-1))
Sine-Cosine Sum/Difference 2	arcsinh(x)±arccosh(y)=±arccosh(y√(x²+1)±x√(y²-1))
Tangent Sum/Difference	arctanh(x)±arctanh(y)=arctanh((x±y)/(1±xy))

(HI) Logarithmic/Compositional Conversions

Sine Logarithmic	ln(x)=arcsinh((x²-1)/2x), x>0
Cosine Logarithmic	ln(x)=±arccosh((x²+1)/2x)
Tangent Logarithmic	ln(x)=arctanh((x²-1)/(x²+1))

Sine Hyperbolic-Circular 1	arcsinh(tan(x))=ln(sec(x+πn)+tan(x+πn))
Sine Hyperbolic-Circular 2	arcsinh(tan(x))=±arccosh(sec(x+πn))
Tangent Hyperbolic-Circular 1	arctanh(cos(2x))=ln(\|cot(x)\|)
Tangent Hyperbolic-Circular 2	±arctanh(sin(x))=±arcsinh(tan(x))

Sine Cofunctional 1	arcsinh(x)=arctanh(x/√(1+x²))
Sine Cofunctional 2	arcsinh(x)=±arccosh(√(1+x²))
Cosine Cofunctional 1	arccosh(x)=\|arcsinh(√(x²-1))\|, x≥1
Cosine Cofunctional 2	arccosh(x)=\|arctanh(√(x²-1)/x)\|, x≥1
Tangent Cofunctional 1	arctanh(x)=arcsinh(x/√(1-x²))
Tangent Cofunctional 2	arctanh(x)=±arccosh(1/√(1-x²))

Unit Hyperbola

x²-y²=1

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Published: 26th August, 2024
Last Updated: 27th August, 2024

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Unit Hyperbola

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