Cheatography

# Trigonometric Properties and Identities Cheat Sheet by CROSSANT

Trigonometry Cheat Sheet for definitions, properties, and identities of Circular & Hyperbolic functions and their inverses. θ, φ, x, and y are all real numbers, with θ and φ in radians. z is a complex variable of the form a+bi, where a and b are real numbers, and i is the imaginary unit. C=Circular, H=Hyperbolic, I=-Inverse

### 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 einθ=(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)=(eθ-e-θ)/2 (-∞,∞) (-∞,∞) Hyperbolic Cosine Function cosh(θ)=(eθ+e-θ)/2 (-∞,∞) [1,∞) Hyperbolic Tangent Function tanh(θ)=(eθ-e-θ)/(eθ+e-θ) (-∞,∞) (-1,1) Hyperbolic Cosecant Function csch(θ­)=2/(eθ-e-θ) (-∞,0)­∪(0,∞) (-∞,0)­∪(0,∞) Hyperbolic Secant Function sech(θ­)=2/(eθ+e-θ) (-∞,∞) (0,1] Hyperbolic Cotangent Function coth(θ)=(eθ+e-θ)/(eθ-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) enθ=(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)=(eiz-e-iz)/2i sin(z)­=-i­sin­h(iz) Complex Cosine cos(z)=(eiz+e-iz)/2 cos(z)­=co­sh(iz) Complex Tangent tan(z)­=-i(eiz-e-iz)/(eiz+e-iz) tan(z)­=-i­tan­h(iz) Complex Cosecant csc(z)­=2i/(eiz-e-iz) csc(z)­=ic­sch(iz) Complex Secant sec(z)­=2/(eiz+e-iz) sec(z)­=se­ch(iz) Complex Cotangent cot(z)=i(eiz+e-iz)/(eiz-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)=∑nk=0 (nk)cosk(θ)sinn-k(θ)sin­((π­/2)­(n-k))
Cosine Multip­le-­Angle Formula: cos(nθ)=∑nk=0 (nk)cosk(θ)sinn-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))

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 AT=½bh units2 Area of a Circle AC=πr2 units2 Pythag­orean Theorem a2+b2=c2
a, b, and c are side lengths of a right-­tri­angle

### (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))

### 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))

x2-y2=1