Trigonometry

All you ever wanted to know about trigonometric functions:
Names and abrevations
Defined from sine or hyperbolics
Derivatives
Defined by right triangle
Odd or even functions
Evaluated in origo
Euler definitions
Idiots formulas (identities)
Half angle formulas
Multiple angle formulas 2x 3x 4x
Integrals
Definitions of inverse hyperbolic functions from natural logarithms
Power formulas x² x³ x⁴
Sums, differences and products
Graphs
Area formulas
Law of Sines
Law of Cosines
Law of Tangents
Special angels
Transhyperbolic relations
Definitions

Names and abrevations:

function hyperbolic inverse (arcus) inverse hyperbolic (area)

sin (sine) sinh sin^-1 (or Arcsin) sinh^-1 (or Arsinh)

cos (cosine) cosh cos^-1 (or Arccos) cosh^-1 (or Arcosh)

tan (tangent) tanh tan^-1 (or Arctan) tanh^-1 (or Artanh)

cot (cotangent) coth cot^-1 (or Arccot) coth^-1 (or Arcoth)

sec (secant) sech sec^-1 (or Arcsec) sech^-1 (or Arsech)

csc (cosecant) csch csc^-1 (or Arccsc) csch^-1 (or Arcsch)

Defined from sine or hyperbolics:
cos(A) = sin(A+π/2)
tan(A) = sin(A)/cos(A) = sin(A)/sin(A+π/2)
cot(A) = cos(A)/sin(A) = sin(A+π/2)/sin(A)
sec(A) = 1/cos(A) = 1/sin(A+π/2) = csc(A+π/2)
csc(A) = 1/sin(A)

tanh(A) = sinh(A)/cosh(A)
coth(A) = cosh(A)/sinh(A)
sech(A) = 1/cosh(A)
csch(A) = 1/sinh(A)

Defined by right triangle:
sin(A) = a/c
cos(A) = b/c
tan(A) = a/b
cot(A) = b/a
sec(A) = c/b
csc(A) = c/a

Euler definitions:

sin(x) = (e^ix-e^-ix)/2i sinh(x) = (e^x-e^-x)/2

cos(x) = (e^ix+e^-ix)/2 cosh(x) = (e^x+e^-x)/2

tan(x) = (e^ix-e^-ix)/i(e^ix+e^-ix) tanh(x) = (e^x-e^-x)/(e^x+e^-x)

cot(x) = i(e^ix+e^-ix)/(e^ix-e^-ix) coth(x) = (e^x+e^-x)/(e^x-e^-x)

sec(x) = 2/(e^ix+e^-ix) sech(x) = 2/(e^x-e^-x)

csc(x) = 2i/(e^ix-e^-ix) csch(x) = 2/(e^x+e^-x)

Definitions of inverse hyperbolic functions from natural logarithms:

sinh^-1(x) = ln (x+(x²+1)^1/2), all x
cosh^-1(x) = ln(x+(x²-1)^1/2), x>=1
tanh^-1(x) = ln((1+x)/(1-x))/2, -1 < x < 1
coth^-1(x) = ln((x+1)/(x-1))/2, 1<x or x<-1
sech^-1(x) = ln(x^-1+(x^-2-1)^1/2), 0<x<=1
csch^-1(x) = ln(x^-1+(x^-2+1)^1/2), x<>0

Algebraic stuff

Sums, differences and products:
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)
cos(A) + sin(A) = &radic(2)sin(A + &pi/4)
cos(A) - sin(A) = &radic(2)cos(A + &pi/4)
sin(A)sin(B) = ½(cos(A - B) - cos(A + B))
cos(A)cos(B) = ½(cos(A - B) + cos(A + B))
sin(A)cos(B) = ½(sin(A - B) + sin(A + B))

Half angle formulas:
sin(x/2) = ±(1/2-cos(x)/2)^1/2
cos(x/2) = ±(1/2+sin(x)/2)^1/2
tan(x/2) = sin(x)/(1+cos(x)) = (1-cos(x))/sin(x) = csc(x) - cot(x)
cot(x/2) = (1+cos(x))/sin(x) = sin(x)/(1-cos(x))
sec(x/2) =
csc(x/2) =

sinh(x/2) = ±(cosh(x)/2-1/2)^1/2
cosh(x/2) = ±(cosh(x)/2+1/2)^1/2
tanh(x/2) = sinh(x)/(1+cosh(x)) = (cosh(x)-1)/sinh(x)
coth(x/2) = (1+cosh(x))/sinh(x) = sinh(x)/(cosh(x)-1)
sech(x/2) =
csch(x/2) =

Multiple angle formulas:
2x:
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)]
sinh(2x) = 2sinh(x)cosh(x)
cosh(2x) = cos²(x) + sin²(x) = 1 + 2sin²(x) = 2cos²(x) - 1
tanh(2x) = 2tanh(x)/[1 + tanh²(x)]
3x:
sin(3x) = 3sin(x) - 4sin³(x)
cos(3x) = 4cos³(x) - 3cos(x)
tan(3x) = [3tan(x) - tan³(x)]/[1 - 3tan²(x)]
sinh(3x) = 3sinh(x) + 4sinh³(x)
cosh(3x) = 4cosh³(x) - 3cosh(x)
tanh(3x) = [3tanh(x) + tanh³(x)]/[1 + 3tanh²(x)]

4x:
sin(4x) = 4sin(x)cos(x) - 8sin³(x)cos(x)
cos(4x) = 8cos⁴(x) - 8cos²(x) + 1
tan(4x) = [4tan(x) - 4tan³(x)]/[1 - 6tan²(x) + tan⁴(x)]
sinh(4x) = 4sinh(x)cosh(x) + 8sinh³(x)cosh(x)
cosh(4x) = 8cosh⁴(x) - 8cosh²(x) + 1
tanh(4x) = [4tanh(x) + 4tanh³(x)]/[1 + 6tanh²(x) + tanh⁴(x)]

Power formulas:
X²:
sin²(x) = [1 - cos(2x)]/2
cos²(x) = [1 + cos(2x)]/2
sinh²(x) = [cosh(2x) - 1]/2
cosh²(x) = [cosh(2x) + 1]/2

X³:
sin³(x) = [3sin(x) - sin(3x)]/4
cos³(x) = [3cos(x) + cos(3x)]/4
sinh³(x) = [sinh(3x) - 3sinh(x)]/4
cosh³(x) = [cosh(3x) + 3cosh(x)]/4

X⁴:
sin⁴(x) = [3 - 4cos(2x) + cosh(4x)]/8
cos⁴(x) = [3 + 4cos(2x) + cosh(4x)]/8
sinh⁴(x) = [3 - 4cosh(2x) + cosh(4x)]/8
cosh⁴(x) = [3 + 4cosh(2x) + cosh(4x)]/8

Other stuff

Idiots formulas (identities):
sin²(x) + cos²(x) = 1
sec²(x) - tan²(x) = 1
csc²(x) - cot²(x) = 1

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

sin^-1(x) + cos^-1(x) = π/2
tan^-1(x) + cot^-1(x) = π/2
sec^-1(x) + csc^-1(x) = π/2

Odd or even functions:

sin(-x) = - sin(x) sinh(-x) = -sinh(x) sin^-1(-x) = -sin^-1(x) sinh^-1(-x) = -sinh^-1(x)

cos(-x) = cos(x) cosh(-x) = cosh(x) cos^-1(-x) = π-cos^-1(x) cosh^-1(-x) =

tan(x) = -tan(x) tanh(-x) = -tanh(x) tan^-1(-x) = -tan^-1(x) tanh^-1(-x) = -tanh^-1(x)

cot(-x) = -cot(x) cot(-x) = -coth(x) cot^-1(-x) = π-cot^-1(x) coth^-1(-x) = -coth^-1(x)

sec(-x) = sec(x) sech(-x) = sech(x) sec^-1(-x) = π-sec^-1(x) sech^-1(-x) =

csc(x) = -csc(x) csch(-x) = -csch(x) csc^-1(-x) = -csc^-1(x) csch^-1(-x) = -csch^-1(x)

Transhyperbolic relations:

sin(x) = -i sinh(ix) sinh(x) = -i sin(ix) sin^-1(x) = -i sinh^-1(ix) sinh^-1(x) = -i sin^-1(ix)

cos(x) = cosh(ix) cosh(x) = cos(ix) cos^-1(x) = ±i cosh^-1(x) cosh^-1(x) = ±i cos^-1(x)

tan(X) = -i tan(ix) tanh(x) = -i tan(ix) tan^-1(x) = -i tanh^-1(ix) tanh^-1(x) = -i tan^-1(ix)

cot(x) = i coth(ix) coth(x) = i cot(ix) cot^-1(x) = i coth^-1(ix) coth^-1(x) = i cot^-1(ix)

sec(x) = sech(ix) sech(x) = sec(ix) sec^-1(x) = ±i sech^-1(x) sech^-1(x) = ±i sec^-1(x)

csc(x) = i csch(ix) csch(x) = i csc(ix) csc^-1(x) = i csch^-1(ix) csch^-1(x) = i csc^-1(ix)

Periodicity:

sin(x + 2n&pi i) = sin(x) sinh(x + 2n&pi i) = sinh(x)

cos(x + 2n&pi i) = cos(x) cosh(x + 2n&pi i) = cosh(x)

tan(x + n&pi i) = tan(x) tanh(x + n&pi i) = tanh(x)

cot(x + n&pi i) = cot(x) coth(x + n&pi i) = coth(x)

sec(x + 2n&pi i) = sec(x) sech(x + 2n&pi i) = sech(x)

csc(x + 2n&pi i) = csc(x) csch(x + 2n&pi i) = csch(x)

Area formulas:
Area = ½(ab sin(C))
Area = ½(ab sin(C))
Area = &radic(s(s - a)(s - b)(s - c)), with s = ½(a + b + c)

Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
bc sin(A) = 2&radic(s(s - a)(s - b)(s - c)), with s = ½(a + b + c)

Law of Cosines:
c² = a² + b² - 2ab cos(C)

Law of Tangents:
(a+b)/(a-b) = tan(½(A + B))/tan(½(A - B))

Dot product:
A·B = AB cos(&theta)

Derivatives and integrals

Derivatives:

d sin(x) = cos(x) dx d sinh(x) = cosh(x) dx d sin^-1 (x) = (1-x²)^-1/2 dx d sinh^-1 (x) = (x²+1)^-1/2 dx

d cos(x) = -sin(x)dx d cosh(x) = sinh(x) dx d cos^-1 (x) = -(1-x²)^-1/2 dx d cosh^-1 (x) = +-(x²-1)^-1/2 dx

d tan (x) = sec²(x) dx d tanh (x) = sech²(x) dx d tan^-1 (x) = (1+x²)^-1 dx d tanh^-1 (x) = (1-x²)^-1 dx

d cot (x) = -csc²(x) dx d coth (x) = -csch²(x) dx d cot^-1 (x) = -(1+x²)^-1 dx d coth^-1 (x) = (1-x²)^-1 dx

d sec (x) = sec(x)tan(x) dx d sech (x) = -sech(x)tanh(x) dx d sec^-1 (x) = (x⁴-x²)^-1/2 dx d sech^-1 (x) = +-(x²-x⁴)^-1/2 dx

d csc (x) = -csc(x)cot(x) dx d csch (x) = -csch(x)coth(x) dx d csc^-1 (x) = -(x⁴-x²)^-1/2 dx d csch^-1 (x) = -(x²+x⁴)^-1/2 dx

Integrals:

∫ sin(x) dx = -cos(x) ∫ sinh(x) dx = cosh(x) ∫ sin^-1(x) dx = xsin^-1(x)+(1-x²)^1/2 ∫ sinh^-1(x) dx = xsinh^-1(x)-(1+x²)^1/2

∫ cos(x) dx = sin(x) ∫ cosh(x) dx = sinh(x) ∫ cos^-1(x) dx = xcos^-1(x)-(1-x²)^1/2 ∫ cosh^-1(x) dx = xcosh^-1(x)+-(x²-1)^1/2

∫ tan(x) dx = -ln cos(x) ∫ tanh(x) dx = ln cosh(x) ∫ tan^-1(x) dx = xtan^-1(x)-(ln(1+x²))/2 ∫ tanh^-1(x) dx = xtanh^-1(x)+(ln(1-x²))/2

∫ cot(x) dx = ln sin(x) ∫ coth(x) dx = ln sinh(x) ∫ cot^-1(x) dx = xcot^-1(x)+(ln(1+x²))/2 ∫ coth^-1(x) dx = xcoth^-1(x)+(ln(x²-1))/2

∫ sec(x) dx = ln (sec(x)+tan(x)) ∫ sech(x) dx = 2tan^-1(x)e^x ∫ sec^-1(x) dx = xsec^-1(x)+-ln(x+(x²-1)^1/2) ∫ sech^-1(x) dx = xsech^-1(x)+-sin^-1(x)

∫ csc(x) dx = ln (csc(x)-cot(x)) ∫ csch(x) dx = ln tanh(x/2) ∫ csc^-1(x) dx = xcsc^-1(x)+-ln(x+(x²-1)^1/2) ∫ csch^-1(x) dx = xcsch^-1(x)+-sinh^-1(x)

Special angles and Graphs

Evaluated in origo:

sin(0) = 0 sinh(0) = 0 sin^-1(0) = 0 sinh^-1(0) = 0

cos(0) = 1 cosh(0) = 1 cos^-1(0) = π/2 cosh^-1(0) = i π/2

tan(0) = 0 tanh(0) = 0 tan^-1(0) = 0 tanh^-1(0) = 0

cot(0) = ∞ cot(0) = ∞ cot^-1(0) = π/2 coth^-1(0) =

sec(0) = 1 sech(0) = 1 sec^-1(0) = sech^-1(0) = ∞

csc(0) = ∞ csch(0) = ∞ csc^-1(0) = csch^-1(0) = ∞

Special angels:

angle in degrees angle in radians sin(x) cos(x) tan(x) cot(x) sec(x) csc(x)

0^o 0 0 1 0 ∞ 1 ∞

15^o π/12 (√6-√2)/4 (√6+√2)/4 2-√3 2+√3 √6-√2 √6+√2

30^o π/6 1/2 √3/2 √3/3 √3 2√3/3 2

45^o π/4 √2/2 √2/2 1 1 √2 √2

60^o π/3 √3/2 1/2 √3 √3/3 2 2√3/3

75^o 5π/12 (√6+√2)/4 (√6-√2)/4 2+√3 2-√3 √6+√2 √6-√2

90^o π/2 1 0 ∞ 0 ∞ 1

105^o 7π/12 (√6+√2)/4 -(√6-√2)/4 -2-√3 -2+√3 -√6-√2 √6-√2

120^o 2π/3 √3/2 -1/2 -√3 -√3/3 -2 2√3/3

135^o 3π/4 √2/2 -√2/2 -1 -1 -√2 √2

150^o 5π/6 1/2 -√3/2 -√3/3 -√3 -2√3/3 2

165^o 11π/12 (√6-√2)/4 -(√6+√2)/4 -2+√3 -2-√3 -√6+√2 √6+√2

180^o π 0 -1 0 ∞ -1 ∞

195^o 13π/12 -(√6-√2)/4 -(√6+√2)/4 2-√3 2+√3 -√6+√2 -√6-√2

210^o 7π/6 -1/2 -√3/2 √3/3 √3 -2√3/3 -2

225^o 5π/4 -√2/2 -√2/2 1 1 -√2 -√2

240^o 4π/3 -√3/2 -1/2 √3 √3/3 -2 - 2√3/3

255^o 17π/12 -(√6+√2)/4 -(√6-√2)/4 2+√3 2-√3 -√6-√2 -√6+√2

270^o 3π/2 -1 0 ∞ 0 ∞ -1

285^o 19π/12 -(√6+√2)/4 (√6-√2)/4 -2-√3 -2+√3 √6+√2 -√6+√2

300^o 5π/3 -√3/2 1/2 -√3 -√3/3 2 - 2√3/3

315^o 7π/4 -√2/2 √2/2 -1 -1 √2 -√2

330^o 11π/6 -1/2 √3/2 -√3/3 -√3 2√3/3 -2

345^o 23π/12 -(√6-√2)/4 (√6+√2)/4 -2+√3 -2-√3 √6-√2 -√6-√2

360^o 2π 0 1 0 ∞ 1 ∞

Graphs:

Home

function	hyperbolic	inverse (arcus)	inverse hyperbolic (area)
sin (sine)	sinh	sin^-1 (or Arcsin)	sinh^-1 (or Arsinh)
cos (cosine)	cosh	cos^-1 (or Arccos)	cosh^-1 (or Arcosh)
tan (tangent)	tanh	tan^-1 (or Arctan)	tanh^-1 (or Artanh)
cot (cotangent)	coth	cot^-1 (or Arccot)	coth^-1 (or Arcoth)
sec (secant)	sech	sec^-1 (or Arcsec)	sech^-1 (or Arsech)
csc (cosecant)	csch	csc^-1 (or Arccsc)	csch^-1 (or Arcsch)

sin(x) = (e^ix-e^-ix)/2i	sinh(x) = (e^x-e^-x)/2
cos(x) = (e^ix+e^-ix)/2	cosh(x) = (e^x+e^-x)/2
tan(x) = (e^ix-e^-ix)/i(e^ix+e^-ix)	tanh(x) = (e^x-e^-x)/(e^x+e^-x)
cot(x) = i(e^ix+e^-ix)/(e^ix-e^-ix)	coth(x) = (e^x+e^-x)/(e^x-e^-x)
sec(x) = 2/(e^ix+e^-ix)	sech(x) = 2/(e^x-e^-x)
csc(x) = 2i/(e^ix-e^-ix)	csch(x) = 2/(e^x+e^-x)

sin(-x) = - sin(x)	sinh(-x) = -sinh(x)	sin^-1(-x) = -sin^-1(x)	sinh^-1(-x) = -sinh^-1(x)
cos(-x) = cos(x)	cosh(-x) = cosh(x)	cos^-1(-x) = π-cos^-1(x)	cosh^-1(-x) =
tan(x) = -tan(x)	tanh(-x) = -tanh(x)	tan^-1(-x) = -tan^-1(x)	tanh^-1(-x) = -tanh^-1(x)
cot(-x) = -cot(x)	cot(-x) = -coth(x)	cot^-1(-x) = π-cot^-1(x)	coth^-1(-x) = -coth^-1(x)
sec(-x) = sec(x)	sech(-x) = sech(x)	sec^-1(-x) = π-sec^-1(x)	sech^-1(-x) =
csc(x) = -csc(x)	csch(-x) = -csch(x)	csc^-1(-x) = -csc^-1(x)	csch^-1(-x) = -csch^-1(x)

sin(x) = -i sinh(ix)	sinh(x) = -i sin(ix)	sin^-1(x) = -i sinh^-1(ix)	sinh^-1(x) = -i sin^-1(ix)
cos(x) = cosh(ix)	cosh(x) = cos(ix)	cos^-1(x) = ±i cosh^-1(x)	cosh^-1(x) = ±i cos^-1(x)
tan(X) = -i tan(ix)	tanh(x) = -i tan(ix)	tan^-1(x) = -i tanh^-1(ix)	tanh^-1(x) = -i tan^-1(ix)
cot(x) = i coth(ix)	coth(x) = i cot(ix)	cot^-1(x) = i coth^-1(ix)	coth^-1(x) = i cot^-1(ix)
sec(x) = sech(ix)	sech(x) = sec(ix)	sec^-1(x) = ±i sech^-1(x)	sech^-1(x) = ±i sec^-1(x)
csc(x) = i csch(ix)	csch(x) = i csc(ix)	csc^-1(x) = i csch^-1(ix)	csch^-1(x) = i csc^-1(ix)

sin(x + 2n&pi i) = sin(x)	sinh(x + 2n&pi i) = sinh(x)
cos(x + 2n&pi i) = cos(x)	cosh(x + 2n&pi i) = cosh(x)
tan(x + n&pi i) = tan(x)	tanh(x + n&pi i) = tanh(x)
cot(x + n&pi i) = cot(x)	coth(x + n&pi i) = coth(x)
sec(x + 2n&pi i) = sec(x)	sech(x + 2n&pi i) = sech(x)
csc(x + 2n&pi i) = csc(x)	csch(x + 2n&pi i) = csch(x)

d sin(x) = cos(x) dx	d sinh(x) = cosh(x) dx	d sin^-1 (x) = (1-x²)^-1/2 dx	d sinh^-1 (x) = (x²+1)^-1/2 dx
d cos(x) = -sin(x)dx	d cosh(x) = sinh(x) dx	d cos^-1 (x) = -(1-x²)^-1/2 dx	d cosh^-1 (x) = +-(x²-1)^-1/2 dx
d tan (x) = sec²(x) dx	d tanh (x) = sech²(x) dx	d tan^-1 (x) = (1+x²)^-1 dx	d tanh^-1 (x) = (1-x²)^-1 dx
d cot (x) = -csc²(x) dx	d coth (x) = -csch²(x) dx	d cot^-1 (x) = -(1+x²)^-1 dx	d coth^-1 (x) = (1-x²)^-1 dx
d sec (x) = sec(x)tan(x) dx	d sech (x) = -sech(x)tanh(x) dx	d sec^-1 (x) = (x⁴-x²)^-1/2 dx	d sech^-1 (x) = +-(x²-x⁴)^-1/2 dx
d csc (x) = -csc(x)cot(x) dx	d csch (x) = -csch(x)coth(x) dx	d csc^-1 (x) = -(x⁴-x²)^-1/2 dx	d csch^-1 (x) = -(x²+x⁴)^-1/2 dx

∫ sin(x) dx = -cos(x)	∫ sinh(x) dx = cosh(x)	∫ sin^-1(x) dx = xsin^-1(x)+(1-x²)^1/2	∫ sinh^-1(x) dx = xsinh^-1(x)-(1+x²)^1/2
∫ cos(x) dx = sin(x)	∫ cosh(x) dx = sinh(x)	∫ cos^-1(x) dx = xcos^-1(x)-(1-x²)^1/2	∫ cosh^-1(x) dx = xcosh^-1(x)+-(x²-1)^1/2
∫ tan(x) dx = -ln cos(x)	∫ tanh(x) dx = ln cosh(x)	∫ tan^-1(x) dx = xtan^-1(x)-(ln(1+x²))/2	∫ tanh^-1(x) dx = xtanh^-1(x)+(ln(1-x²))/2
∫ cot(x) dx = ln sin(x)	∫ coth(x) dx = ln sinh(x)	∫ cot^-1(x) dx = xcot^-1(x)+(ln(1+x²))/2	∫ coth^-1(x) dx = xcoth^-1(x)+(ln(x²-1))/2
∫ sec(x) dx = ln (sec(x)+tan(x))	∫ sech(x) dx = 2tan^-1(x)e^x	∫ sec^-1(x) dx = xsec^-1(x)+-ln(x+(x²-1)^1/2)	∫ sech^-1(x) dx = xsech^-1(x)+-sin^-1(x)
∫ csc(x) dx = ln (csc(x)-cot(x))	∫ csch(x) dx = ln tanh(x/2)	∫ csc^-1(x) dx = xcsc^-1(x)+-ln(x+(x²-1)^1/2)	∫ csch^-1(x) dx = xcsch^-1(x)+-sinh^-1(x)

sin(0) = 0	sinh(0) = 0	sin^-1(0) = 0	sinh^-1(0) = 0
cos(0) = 1	cosh(0) = 1	cos^-1(0) = π/2	cosh^-1(0) = i π/2
tan(0) = 0	tanh(0) = 0	tan^-1(0) = 0	tanh^-1(0) = 0
cot(0) = ∞	cot(0) = ∞	cot^-1(0) = π/2	coth^-1(0) =
sec(0) = 1	sech(0) = 1	sec^-1(0) =	sech^-1(0) = ∞
csc(0) = ∞	csch(0) = ∞	csc^-1(0) =	csch^-1(0) = ∞

angle in degrees	angle in radians	sin(x)	cos(x)	tan(x)	cot(x)	sec(x)	csc(x)
0^o	0	0	1	0	∞	1	∞
15^o	π/12	(√6-√2)/4	(√6+√2)/4	2-√3	2+√3	√6-√2	√6+√2
30^o	π/6	1/2	√3/2	√3/3	√3	2√3/3	2
45^o	π/4	√2/2	√2/2	1	1	√2	√2
60^o	π/3	√3/2	1/2	√3	√3/3	2	2√3/3
75^o	5π/12	(√6+√2)/4	(√6-√2)/4	2+√3	2-√3	√6+√2	√6-√2
90^o	π/2	1	0	∞	0	∞	1
105^o	7π/12	(√6+√2)/4	-(√6-√2)/4	-2-√3	-2+√3	-√6-√2	√6-√2
120^o	2π/3	√3/2	-1/2	-√3	-√3/3	-2	2√3/3
135^o	3π/4	√2/2	-√2/2	-1	-1	-√2	√2
150^o	5π/6	1/2	-√3/2	-√3/3	-√3	-2√3/3	2
165^o	11π/12	(√6-√2)/4	-(√6+√2)/4	-2+√3	-2-√3	-√6+√2	√6+√2
180^o	π	0	-1	0	∞	-1	∞
195^o	13π/12	-(√6-√2)/4	-(√6+√2)/4	2-√3	2+√3	-√6+√2	-√6-√2
210^o	7π/6	-1/2	-√3/2	√3/3	√3	-2√3/3	-2
225^o	5π/4	-√2/2	-√2/2	1	1	-√2	-√2
240^o	4π/3	-√3/2	-1/2	√3	√3/3	-2	- 2√3/3
255^o	17π/12	-(√6+√2)/4	-(√6-√2)/4	2+√3	2-√3	-√6-√2	-√6+√2
270^o	3π/2	-1	0	∞	0	∞	-1
285^o	19π/12	-(√6+√2)/4	(√6-√2)/4	-2-√3	-2+√3	√6+√2	-√6+√2
300^o	5π/3	-√3/2	1/2	-√3	-√3/3	2	- 2√3/3
315^o	7π/4	-√2/2	√2/2	-1	-1	√2	-√2
330^o	11π/6	-1/2	√3/2	-√3/3	-√3	2√3/3	-2
345^o	23π/12	-(√6-√2)/4	(√6+√2)/4	-2+√3	-2-√3	√6-√2	-√6-√2
360^o	2π	0	1	0	∞	1	∞