-
Definition of the Six Trigonometric Functions Right triangle
definitions, where 0 < < / 2.
.
.
tan
.
.
cos
.
.
sin
adjopphypadjhypopp
=
=
=
.
.
cot
.
.
sec
.
.
csc
oppadjadjhypopphyp
=
=
=
Circular Function Definitions, where is any angle
x
yr
x
r
y
=
=
=
tan
cos
sin
yx
x
r
yr
=
=
=
cot
sec
csc
Units Circle
707.02
122
== 866.023
= 500.021
=
Common Reference Triangles
Reciprocal Identities
xx
xx
tan1
cot
cot1
tan
=
=
xx
xx
sec
1cos
csc
1sin
=
=
xx
xx
cos
1sec
sin1
csc
=
=
Tangent and Cotangent Identities
x
xx
cos
sintan =
x
xx
sincos
cot =
Hyperbolic Functions
2sinh
xx eex
=
2cosh
xx eex
+=
x
xx
coshsinh
tanh =
xx eexhx
==
2sinh
1csc
xx eexhx
+==
2cosh
1sec
x
x
xx
sinhcosh
tanh1
coth ==
Sum and Difference Formulas ( )( ) vuvuvu
vuvuvu
sinsincoscoscossincoscossinsin
=
=
( )vu
vuvu
tantan1tantan
tan
=
Double-Angle Formulas
uuu cossin22sin = u
uu 2tan1
tan22tan
=
uuuuu 2222 sin211cos2sincos2cos === Power Reducing Formulas
( )uuu 2cos121
2cos21
sin 2 == u
uu
2cos12cos1
tan 2+
=
( )uuu 2cos121
2cos21
cos2 +=+
=
x
x
x
xx
2cos12sin
2sin2cos1
tan+
=
=
Sum-to-Product Formulas
+=
+=+
+=
+=+
2sin
2sin2coscos
2cos
2cos2coscos
2sin
2cos2sinsin
2cos
2sin2sinsin
vuvuvu
vuvuvu
vuvuvu
vuvuvu
Reduction Formulas ( )( ) xx
xx
csccsc
sinsin=
=
( )( ) xx
xx
tantan
coscos
=
=
( )( ) xx
xx
cotcot
secsec
=
=
Pythagorean Identities 1cossin 22 =+ xx xx 22 sectan1 =+ xx 22
csccot1 =+
-
Fractions
bdac
dc
ba
bdbcad
dc
ba
=
+=+
bcad
c
dba
dcba
==
ba
ba
ba
==
( ) ( )( ) ( )
( ) ( )( ) ( )
( )( )bdfgeh
fhbcadbdfhbdfh
hgfedcba
hgfedcba
+
+=
+
+=
+
+
////
////
Exponents
m
m
nm
n
m
mnmn
AA
AAA
AAA
1=
=
=
+
( )( )
n
nn
nnn
mnnm
BA
BA
BAAB
AA
=
=
=
( )0102/1
/
=
=
=
AAAA
AA n mnm
Logarithms log = common logarithm (logarithm to the base 10) =
10log ln = natural logarithm (logarithm to the base e) = elog e =
2.71828182846
AeBA
A
A
B
=
=
=
lnlog10
ABA
Ae
A
B
=
=
=
10logln
01ln1log10 log
ln
==
=
=
AAe
A
A
xBAe
=
=
=
1ln110log
BABA
BAAB
logloglog
logloglog
==
==
AnA
AA
n loglog
log1log
=
=
BA
xloglog
=
( )( ) AAA log30258509.2log10lnln == ( )( ) AAeA
ln43429448.0lnloglog ==
Law of Sines Law of Cosines
Cc
Bb
Aa
sinsinsin== Cabbac cos2222 +=
CabBacAbcArea sin21
sin21
sin21
===
=++ 180CBA
Similar Triangles
c
c
bb
a
a '''==
Product to Sum Formulas
( ) ( )[ ]( ) ( )[ ]
( ) ( )[ ]( ) ( )[ ]vuvuvu
vuvuvu
vuvuvu
vuvuvu
+=
++=
++=
+=
sinsin21
sincos
sinsin21
cossin
coscos21
coscos
coscos21
sinsin
Cofunction Identities
xx
xx
xx
csc2
sec
sec2
csc
cos2
sin
=
=
=
pi
pi
pi
xx
xx
xx
tan2
cot
cot2
tan
sin2
cos
=
=
=
pi
pi
pi
xx
xx
xx
cos2
sin
sin2
cos
cos2
sin
=
+
=
=
pi
pi
pi
2cos1
2sin
2cos1
2cos
sin2
cos
xx
xx
xx
=
+=
=
+
pi
x
xx
sincos1
2tan
=
Quadradic Equation and Quadradic Formula
)0(02
=++
a
cbxax
a
acbbx
242
=
Temperature Conversion Formulas
( )
( )( ) 67.459
5915.273
59
15.2739515.27332
95
67.45967.4595932
59
15.27315.2739532
95
15.273067.4590
+==+=
+==+=
==+=
===
==
FKCR
CRFK
RKCF
KRFC
KCRF
Distance Between Points ( )11 , yxP and ( )22 , yxQ is,
( ) ( ) ( )221221, yyxxQPd ++= Slope of Line Between Points
12
12
xx
yym
=
Distance Between any Point ( )11 , yxP and any line 0=++ CBxAx
is,
22
11
BA
CByAxd
+
++=
-
Factoring Formulas ( )( )
( ) 22222
2 yxyxyx
yxyxyx
+=
+= ( )( )( ) 32233
2233
33 yxyyxxyxyxyxyxyx
+=
+=
The Greek Alphabet Alpha Beta Gamma Delta Epsilon Zeta Eta
Theta
Iota Kappa Lambda Mu Nu Xi Omicron Pi pi
Rho Sigma Tau Upsilon Phi Chi Psi Omega
Binomial Theorem For any positive integer n,
( ) ( )( )( ) nnn
nnnn
bnabbannn
bannbnaaba
+++
+
++=+
133
221
...
32121
211
For instance,
( )( )( )( ) 4322344
32233
222
1
46433
2
babbabaabababbaaba
bababababa
++++=+
+++=+
++=+
+=+
Difference of Like Integers Powers, n > 1 ( )( )122321 ...
+++++= nnnnnnn babbabaababa
For instance, ( )( )( )( )( )( )322344
2233
22
babbaababababababa
bababa
+++=
++=
+=
Series Formulas
...
!7!5!3sin
753
++=xxx
xx ...!6!4!2
1cos642
++=xxx
x
...
!7!5!3sinh
753
++++=xxx
xx ...!6!4!2
1cosh642
++++=xxx
x
...
!4!3!21
432
+++++=xxx
xe x
( )11...,753
tan753
1 ++= xxxxxx
( )11...,11
1 432
-
Triangle
bhArea
ah
21
sin
=
=
Law of Cosines cos2222 abbac +=
Right Triangle
Pythagorean Theorem 222 bac +=
Equilateral Triangle
243
23
2 shsArea
sh
==
=
32h
s =
Parallelogram
bhArea =
Trapezoid
( )bahArea +=2
Circle
rDnceCircumfere
DDrArea
DDr
rD
pipi
pipipi
242
2212
222
==
=
==
==
=
Sphere
2
3
434
rAreaSurfacerVolume
pi
pi
=
=
Circular Ring
( p = average radius, w = width of ring) ( )22 rRArea = pi
pwpi2=
Sector of a Circle ( in Radians)
22
2
2
2
==
DrArea
28
2rsD
==
2D
rs ==
Sector of a Circular Ring ( p = average radius, w = width of
ring, in Radians)
pwArea =
Ellipse
abcVolumeEllipsoid
banceCircumfere
abArea
pi
pi
pi
43
22
22
=
+
=
Right Circular Cone
3
2hrVolume pi=
Lateral Surface Area =
rshrr pipi =+ 22
Any Cone or Pyramid
(A = Area of Base)
3AhVolume =
-
Right Circular Cylinder
hrVolume 2pi=
Lateral Surface Area = rhpi2
Wedge
(A = area of upper face, B = area of base)
secBA =
Frustum of Right Circular Cone
( )3
22 hRrRrVolume ++= pi
Lateral Surface Area =
( )rRs +pi
Any Cylinder or Prism with Parallel Bases
bhV =
b = area of base
Prefixes for Powers of Ten Power
24
21
18
15
12
9
6
3
2
1
0
1
2
3
6
9
12
15
18
21
24
101010101010101010101010101010101010101010
Scientific 1E-24 1E-21 1E-18 1E-15 1E-12 1E-9 1E-6 1E-3 1E-2
1E-1 1E0 1E1 1E2 1E3 1E6 1E9 1E12 1E15 1E18 1E21 1E24
Actual Number 0.000,000,000,000,000,000,000,000,1
0.000,000,000,000,000,000,000,1 0.000,000,000,000,000,001
0.000,000,000,000,001 0.000,000,000,001 0.000,000,001 0.000,001
0.001 0.01 0.1 1 10 100 1,000 1,000,000 1,000,000,000
1,000,000,000,000 1,000,000,000,000,000 1,000,000,000,000,000,000
1,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000
Abbrev. y z
a
f p n
m
c
d ---
D h or H k or K M G T P E Z Y
Name yocto zepto atto
femto pico nano
micro milli centi deci ---
deka hecto kilo mega giga tera
peta exa
zeta
yotta
-
Some Fundamental Constants **Numbers in parenthesis represent
how many total decimal places it is carried out to be nearly
exact.
Atomic Mass Unit u 1.660 540 2(10) x 2710 kg 931.494 32(28) MeV/
2c Avogadros Number AN 6.022 136 7(36) x 2310 1)( mol Average
Earth-Moon Distance 3.84 x 810 m 1.259 842 519 690 x 910 ft Average
Earth-Sun Distance 1.496 x 1110 m 4.908 136 482 940 x 1110 ft
Average Radius of the Earth 6.37 x 610 m 2.089 895 013 120 x 710
ft
Bohr Magneton e
Bm
e
2
= 9.274 015 4(31) x 2410 J/T
Bohr Radius ee kem
a 2
2
0
= 0.529 177 249(24) x 1010 m
Boltzmanns Constant A
B NRk = 1.380 658(12) x 2310 J/K
7.270 063 x 2710 Btu/R
Compton Wavelength cm
he
C = 2.426 310 58(22) x 1210 m
Density of Air (20C and 1 atm) 1.20 kg/ 3m 0.074 913 600 lbm/
3ft Density of Water (20C and 1 atm) 1.00 x 310 kg/ 3m 62.428 lbm/
3ft Deuteron Mass dm 3.343 586 0(20) x 2710 kg 2.013 533 214(24) u
Diracs Constant
pi2h
= 1.054 572 66(63) x 3410 Js 2.77 505 62 x 4110 Btuhr Electron
Mass em 9.109 389 7(54) x 3110 kg 5.485 799 03(13) x 410 u 0.510
999 06(15) MeV/ 2c Electron-Volt eV 1.602 177 33(49) x 1910 J
Elementary Charge e 1.602 177 33(49) x 1910 C Faradays Constant F
964785.309 C/gmol Gas Constant R 8.314 510(70) J/Kmol 10.731 640
psia/ 3ft Gravitational Constant G 6.672 59(85) x 1110 N 2m / 2kg
3.322 80 x 1110 lbf 22 / lbft Gravity g 9.806 650 m/ 2s (Free-Fall
Acceleration) 32.174 ft/ 2s Hydrogen Ground State
0
2
2
24
0 22 akekem
E eee ==
18.605 98(40) eV
Josephson Frequency- he /2 4.835 976 7(14) x 1410 Hz/V voltage
Ratio
-
Magnetic Flux Quantum e
h20
= 2.067 834 61(61) x 1510 Wb Mass of the Earth 5.98 x 2410 kg
Mass of the Moon 7.36 x 2210 kg Mass of the Sun 1.99 x 3010 kg
Neutron Mass nm 1.674 928 6(10) x 2710 kg 1.008 664 904(14) u
939.565 63(28) MeV/ 2c Nuclear Magneton
pn
m
e
2
= 5.050 786 6(17) x 2710 J/T
Permeability of Free Space 0 4pi x 710 N/ 2A (exact)
Permittivity of Free Space 2
00
1c
= 8.854 187 817 x 2710 2C /N 2m (exact)
Photon Wavelength 0 1239.8425 nm
12398.425
A Photon Frequency 0f 2.417 988 3 x 1410 Hz Plancks Constant h
6.626 075(40) x 3410 Js 1.744 529 9 x 4010 Btuhr Proton Mass pm
1.672 623(10) x 2710 kg 1.007 276 470(12) u 938.272 3(28) MeV/ 2c
Quantized Hall Resistance 2e
h 25812.805 6(12)
Rydberg Constant HR 1.097 373 153 4(13) x 710 1m Speed of Light
in Vacuum c 2.997 924 58 x 810 m/s (exact) 9.835 710 56 x 810 ft/s
Standard Atmospheric Pressure 1.013 x 510 pa 14.692 322 832 psi
Stefan-Boltzmann Constant 5.670 510 x 810 W/ 42 Km 1.712 340 x 910
Btu/h 2ft 4R Weins Constant 3c 0.002 897 756 mK 5215.961 R
CONVERSIONS
LENGTH m = Meter cm = Centimeter mm = Millimeter yd = Yard ft =
Feet (foot) in = Inch Mpc = Megaparsec pc = Parsec lyr = Light Year
au = Astronomical Unit km = Kilometer mi = Mile (International) nmi
= Nautical Mile miUS = US Statute Mile chain = Chain rd = Rod fath
= Fathom ftUS = US Survey Foot mil = Mil = Micron A = Angstrom
fermi = Fermi
1 m = 100 cm = 1000 mm = 1.0936133 yd = 3.28084 ft = 39.37008 in
= 0.001 km = 0.00062137 mi = 0.19884 rd = 0.4568056 fath = 3.280833
ftUS = 1000,000 = 10,000,000,000 A = 1,000,000,000,000,000 fermi 1
cm = 0.01 m = 10 mm = 0.328084 ft = 0.393700787 in = 0.00001 km =
0.000006214 mi = 0.0004971 chain = 0.0019884 rd = 393.700787 mil =
10,000 = 100,000,000 A = 10,000,000,000,000 fermi 1 mm = 0.001 m =
0.1 cm = 0.00328084 ft = 0.03937008 in = 0.000001 km =
0.00000062137 mi = 0.00004971 chain = 0.00019884 rd = 39.3700787
mil = 1000 = 1,000,000 A = 1,000,000,000,000 fermi 1 yd = 0.9144 m
= 91.44 cm = 914.4 mm = 3 ft = 36 in = 0.0009144 km = 0.000568182
mi = 0.181818 rd = 0.5 fath = 2.999994 ftUS = 36,000 mil = 914400 =
9,144,000,000 A = 914,400,000,000,000 fermi 1 ft = 0.3048 m = 30.48
cm = 304.3 mm = 0.3333 yd = 12 in =0.0003048 km = 0.0001893939 mi =
0.060606 fd = 0.16667 fath = 0.999998 ftUS = 12,000 mil = 304,800 =
3,048,000,000 A = 304,800,000,000,000 fermi
-
1 in = 0.0254 m = 2.54 cm = 25.4 mm = 0.02778 yd = 0.083333 ft =
0.0000254 km = 0.00001578383 mi = 0.0050505 rd = 0.01389 fath =
1000 mil = 25,400 = 254,000,000 A = 25,400,000,000,000 fermi 1 Mpc
= 1,000,000 pc = 3,261,634.07982 lyr = 206,264,806,247 au =
3.086E19 km = 1.9174E19 mi 1 pc = 0.000001 Mpc = 3.26163497 lyr =
206,264.806247 au = 3.086E13 km = 1.9174E13 mi 1 lyr = 9.46053E15 m
= 1.034616E16 yd = 3.10385E16 ft = 0.306594785168 pc =
63,239.713959 au = 9.46053E12 km = 5.8785E12 mi 1 au =
149,597,900,000 m = 290,806,758,530 ft = 0.00000485 pc =
149,597,900 km = 92,955,825.4791 mi 1 km = 1000 m = 100,000 cm =
1,000,000 mm = 1093.61329834 yd = 3280.83989502 ft = 39,370.0787402
in = 3.24078E-14 pc = 1.057E-13 lyr = 6.6846E-9 au = 0.62137119224
mi = 0.539956803458 nmi = 0.621369949498 miUS = 49.7095959596 chain
= 198.838383 rd = 546.80556 fath = 3280.83333 ftUS 1 mi = 1609.344
m = 160,934.4 cm = 1,609,344 mm = 1760 yd = 5280 ft = 63,360 in =
5.21553E-14 pc = 1.70111E-13 lyr = 1.07578E-8 au = 1.609344 km =
0.868976241901 nmi = 0.999998 miUS = 79.99984 chain = 319.99936 rd
= 879 99824 fath = 5279.98944 ftUS 1 nmi = 1852 m = 2025.37182852
yd = 6076.11548556 ft = 1.852 km = 1.15077944802 nmi =
1.15077714646 miUS = 92.0621717168 chain = 368.248686867 rd =
1012.68388888 fath = 6076.10333328 ftUS 1 miUS = 1609.34721869 m =
1760.00352 yd = 5280.01056 ft = 1.60935 km = 1.000002 mi = 79.99999
chain = 319.99999 rd = 879.99999 fath 1 chain = 20.1168402337 m =
2011.68402337 cm = 20,116.8402337 mm = 22.000044 yd = 66.000132 ft
= 792.001584 in = 0.02011684 km = 0.0125 mi = 4 rd = 11 fath = 66
ftUS = 792,001.584003 mil 1 rd = 5.02921 m = 502.921 cm = 5029.21
mm = 5.500011 yd = 16.500033 ft = 198.000396 in = 0.00502921 km =
0.003125 mi = 0.25 chain = 2.75 fath = 16.5 ftUS = 198000.396 mil 1
fath = 1.8288 m = 2.000004 yd = 6.000012 ft = 0.0018288 km =
0.00136366 mi = 0.090909 chain = 0.363636 rd = 6 ftUS = 72,000.144
mil 1 ftUS = 0.3048 m = 0.333334 yd = 1.000002 ft = 12.000024 in =
0.0003048 km = 0.000189394 mi = 0.0151515 chain = 0.060606 rd =
12,000.024 mil 1 mil = 0.0000254 m = 0.00254 cm = 0.0254 mm =
0.00008333 ft = 0.001 in = 24.5 = 254,000 A = 25,400,000,000 fermi
1 = 0.000001 m = 0.0001 cm = 0.001 mm = 0.00000328084 ft =
0.00003937 in = 0.03937 mil = 10,000 A = 1,000,000,000 fermi 1 A =
0.0000000001 m = 0.00000001 cm = 0.0000001 mm = 3.937E-9 in =
0.000003937 mil = 0.0001 = 100,000 fermi 1 fermi = 1E-15 m = 1E-13
cm = 1E-12 mm = 3.937E-14 in = 3.937E-11 mil = 0.000000001 =
0.00001 A
AREA m = Meters Squared cm = Centimeters Squared B = Barn yd =
Yard squared ft = Feet squared in = Inches Squared km = Kilometers
Squared Ha = Hectar A = Are mi = Miles squared Acre = Acre miUS =
US Statute Miles Squared
1 m = 10,000 cm = 1.19599 yd = 10.7639 ft = 1550.0031 in =
3.86102E-7 mi = 2.471044E-4 Acre = 1E28 B 1 cm = 0.155 in =
1.0764E-3 ft 1 yd = 0.83613 m = 9 ft = 1296 in 1 ft = 0.11111 yd =
144 in = 929.0304 cm = 0.092903 m 1 in = 6.4516 cm = 6.9444E-3 ft =
0.00064516 m 1 km = 100 Ha = 10,000 A = 247.1044 Acre = 0.386102 mi
1 Ha = 0.01 km = 100 A = 2.471044 Acre = 3.86102E-3 mi 1 A =
2.471044E-4 Acre = 0.01 Ha = 0.0001 km = 3.86102E-5 mi = 1E30 B 1
mi = 2.589988 km = 639.99744 Acre = 0.999996 miUS = 3,097,600 yd =
27,878,400 ft = 2,589,933.11034 m = 258.998811 Ha 1 miUS = 1.000004
mi 1 Acre = 4046.8726 m = 4840.01936 yd = 43,560.17424 ft =
2.5625E-3 mi = 4.046873E-3 km = 0.4046873 Ha = 40.46873 A 1 B =
1E-28 m
VOLUME m = Meters Cubed st = Stere cm = Centimeters Cubed yd =
Yards Cubed \ft = Feet Cubed in = Inches Cubed L = Liter galUK =
Gallon (UK) galC = Gallon (Canadian) gal = Gallon (US) pt = Pint ml
= Milliliter cu = Cup (US) oz = Fluid Ounce (US) ozUK = Fluid Ounce
(UK) tbsp = Tablespoon tsp = Teaspoon bbl = Barrel bu = Bushel pk =
Peck fbm = Board Foot qt = Quart
1 m = 1 st = 1,000,000 cm = 1.30795 yd = 35.314667 ft =
61023.7440946 in = 1000 L = 1,000,000 ml = 264.172 gal = 1056.68821
qt = 2113.37642 pt = 4226.7528377 cu = 33,814.0227 oz = 6.28981 bbl
= 28.3775934 bu = 113.510374 pk 1 st = 1 m (same as above) 1 cm =
0.000001 m = 0.000035315 ft = 0.061024 in = 1000 m = 0.000264172
gal = 0.0010567 qt = 0.0021134 pt = 1 ml = 0.004226753 cu
= 0.033814 oz = 0.067628 tbsp = 0.202884 tsp 1 yd = 0.76455486 m
= 764,554.857984 cm = 27 ft = 46,656 in = 764.55486 L = 201.974026
gal = 807.8961 qt = 1615.79221 pt = 3231.584416 cu = 25,852.67532
oz = 4.808905 bbl = 21.696227 bu = 86.784907 pk 1 ft = 0.02832 m =
28,316.8466 cm = 0.037037 yd = 1728 in = 28.31685 L = 7.48052 gal =
576/77 gal = 29.92208 qt = 59.844156 pt = 119.68831 cu = 957.50649
oz = 0.1781076 bbl = 0.803564 bu = 3.214256 pk = 12 fbm 1 in =
16.387064 cm = 0.000578704 ft = 0.016387 L = 0.004329 gal =
0.017316 qt = 0.034632 pt = 16.387064 ml = 0.069264 cu = 0.5541125
oz = 1.108225 tbsp = 3.32467 tsp 1 L = 0.001 m = 1000 cm =
61.023744 in = 0.264172 gal = 1.0566882 qt = 2.113376 pt = 1000 ml
= 4.226753 cu = 33.814 oz = 67.628 tbsp
= 202.88414 tsp 1 gal UK = 1.00000044 gal C = 1.2009505 gal 1
gal C = 0.99999956 gal UK = 1.2009499 gal 1 gal = 0.83267381 gal UK
= 0.8326742 gal C = 0.003785 m = 3,785.411784 cm = 0.13368055 ft =
231 in = 3.7854 L = 4 qt = 8 pt = 16 cu = 128 oz
= 256 tbsp = 768 tsp = 3,785.411784 ml = 0.0238952 bbl =
0.1074209 bu = 0.4296835 pk 1 qt = 0.25 gal = 2 pt = 4 cu = 32 oz =
64 tbsp = 192 tsp = 0.00946353 m = 946.352946 cm = 57.75 in =
0.946352946 L = 946.352946 cm
= 0.02685522 bu = 0.107451 pk 1 pt = 0.125 gal = 0.5 qt = 2 cu =
16 oz = 32 tbsp = 96 tsp = 473.176473 cm = 28.875 in = 0.473176 L =
473.176473 ml = 0.0134276 bu = 0.05371 pk 1 ml = 1 cm (see 1 cm
above) 1 cu = 8 oz = 16 tbsp = 48 tsp = 0.25 qt = 0.5 pt = 0.0625
gal = 236.5882365 cm = 236.5882365 ml = 14.4375 in = 0.23658824
L
-
1 oz = 1.0408423 ozUK = 2 tbsp = 6 tsp = 29.57353 ml = 29.57353
cm = 0.029574 L = 0.0078125 gal = 0.03125 qt = 0.0625 pt =
1.8046875 in 1 ozUK = 0.96076036 oz 1 tbsp = 3 tsp = 0.5 oz =
0.0625 cu = 14.7867648 ml = 14.7867648 cm = 0.0147868 L =
0.00390625 gal = 0.015625 qt = 0.03125 pt = 0.90234375 in 1 tsp =
0.333333 tbsp = 0.1666667 oz = 0.020833334 cu = 4.9289216 ml =
4.9289216 cm = 0.00492892 L = 0.00130208 gal = 0.0104167 pt =
0.3008 in 1 bbl = 4.51168 bu = 18.0467072 pk = 67.375 fbm =
0.1589873 m = 5.61458334 ft = 0.20794753 yd = 42 gal = 158.987295 L
= 168 qt = 336 pt
= 672 cu = 5376 oz = 10,752 tbsp = 32,256 tsp 1 bu = 0.221647
bbl = 4 pk = 14.93347 fbm = 0.03523907 m = 1.244456 ft = 0.046091
yd = 9.309177 gal = 35.23907 L = 37.23671 qt = 74.4734196 pt 1 pk =
0.25 bu = 0.05541177 bbl = 3.733368 fbm = 0.0088977 m = 0.311114 ft
= 0.0115227 yd = 2.3272944 gal = 8.8097675 L = 9.3091774 qt 1 fbm =
0.2678546 pk = 0.066964 bu = 0.0148423 bbl = 0.00235074 m =
0.083334 ft = 0.00308642 yd = 0.6233766 gal = 2.35974 L = 2.4935065
qt
TIME yr = Year day = Day hr = Hour min = Minute s = Second
1 yr = 365.242198781 day = 8765.81277074 hr = 525,948.766244 min
= 31,556,925.9746 s 1 yr = 365 day = 8760 hr = 525,600 min =
31,536,000 s 1 day = 24 hr = 1400 min = 86,400 s 1 hr = 60 min =
3600 s
VELOCITY ( SPEED ) m/s = Meters per Second ft/s = Feet per
Second kph = Kilometers per Hour cm/s = Centimeters per Second mph
= Miles per Hour c = Speed of Light mi/m = Miles per Minute knot =
Nautical Miles per Hour
1 m/s = 100 cm/s = 3.28084 ft/s = 3.6 kph =2.23693629 mph =
1.9438445 knot 1 cm/s = 0.01 m/s = 3.28084E-2 ft/s = 0.036 kph =
2.23693629E-2 mph = 1.9438445E-2 knot 1 ft/s = 0.3048 m/s = 30.48
m/s = 1.09728 kph = .68181818 mph = .5924838 knot 1 kph = .277778
m/s = 27.77778 cm/s = .9113444 ft/s = 0.62137119 mph = 0.5399568
knot 1 mph = 0.44704 m/s = 44.704 cm/s = 1.466667 ft/s = 22/15 ft/s
= 1.609344 kph = 0.86897624 knot 1 knot = 0.51444 m/s = 51.4444
cm/s = 1.68781 ft/s = 1.852 kph = 1.14077945 mph 1 c = 299,792,458
m/s = 983,571,056.43 ft/s = 1,079,252,848.8 kph = 670,616,629.384
mph = 582,749,918.358 knot 1 mi/m = 26.8224 m/s = 60 mph = 88 ft/s
= 52.1386 knot
ACCELERATION m/s = Meters per Second Squared ft/s = Feet per
Second Squared cm/s = Centimeters per Second Squared
1 m/s = 3.280835 ft/s = 100 cm/s 1 ft/s = 0.3048 m/s = 30.48
cm/s 1 cm/s = 0.01 m/s 9.80665 m/s = 32.174 ft/s
MASS kg = Kilogram g = Gram lb = lbm = Pound (mass) oz = Ounce
slug = Slug lbt = Troy Pound ton = Short Ton tonUK = Long Ton UK T
= Metric Ton ozt = Troy Ounce ct = Carat grain = Grain U = Unified
Atomic Mass
1kg = 1000 g = 2.20462 lb = 35.2739 oz = 0.06852 slug = 0.001 T
= 5000 ct = 6.02E26 U 1 g = 0.001 kg = 0.03527 oz = 0.000001 T = 5
ct = 15.4324 grain 1 lb = 0.45359 kg = 435.592 g = 16 oz = 0.03108
slug = 1.21528 lbt = 0.0005 ton = 0.0004536 T = 14.5833 ozt = 7000
grain = 2.73E26 U 1 oz = 0.02835 kg = 28.3495 g = 0.91145 ozt =
141.747 ct = 437.50 grain
1 slug = 1ftslb 2
= 14.5939 kg = 32.174 lb = 514.785 oz = 469.205 ozt
1 lbt = 373.242 g = 0.82286 lb = 13.1657 oz = 12 ozt = 5760
grain 1 ton = 907.185 kg = 2000 lb = 32000 oz = 14,000,000 grain 1
tonUK = 1016.05 kg = 2240 lb = 35,840 oz = 1.12 ton = 1.01605 T =
6.02E29 U 1 T = 1000 kg = 2204.62 lb = 1.10231 ton = 5,000,000 ct 1
ozt = 31.1035 g = 1.09714 oz = 155.517 ct = 480 grain 1 ct = 0.2 g
= 0.0004409 lb = 0.00705 oz = 0.0000002 T = 3.06865 grain 1 grain =
0.06479 g = 0.02286 oz = 0.32399 ct 1 U = 1.6E-27 kg = 3.66E-27 lb
= 1.6E-30 T = 931.5
2c
MeV
FORCE N = Newton dyn = Dyne lbf = Pound (force) gf = Gram
(force) kip = kiloPound (force) pdl = Poundal
1 N = 12s
mkg = 1E5 dyn = 0.2248 lbf = 101.97162 gf = 7.233 pdl
1 dyn = 1E-5 N = 2.2481E-6 lbf 1 lbf = 4.448222 N =
444822.161526 dyn = 9.001 kip 1 gf = 0.00980665 N 1 kip = 1000
lbf
-
1 pdl = 0.138254954 N
ENERGY, WORK, HEAT J = Joule erg = Erg Kcal = Cal = kilo-Calorie
cal = Calorie Btu = British Thermal Unit ftlb = Foot Pound therm =
Therm (EEC) MeV = Mega electron Volt eV = electron Volt kWh =
kiloWatt hour
1 J = 1 Nm = 1 Ws = 1 2
2
s
mkg = 10,000,000 erg = 0.0002388 Kcal = 0.238846 cal =
0.00094782 Btu = 0.737562 ftlb = 9.47813E-9 therm
= 6.2415E18 eV = 2. 777777778E-7 kWh 1 erg = 0.0000001 J =
624150.636307 MeV = 624,150,636,307 eV 1 Kcal = 4186.8 J =
41,868,000,000 erg = 1000 cal = 3.968321 Btu = 3088.0252066 ftlb =
0.00003968 therm 1 cal = 4.1868 J = 41,868,000 erg = 0.0.001 Kcal =
0.00396832 Btu = 3.0880252 ftlb = 3.9683E-8 therm 1 Btu =
1055.05585 J = 0.251996 Kcal = 251.99576 cal = 778.16926 ftlb =
0.000010002 therm = 0.000293071 kWh 1 ftlb = 1.35582 J = 0.00032383
Kcal = 0.32383155 cal = 0.00128507 Btu = 3.766161E-7 kWh 1 therm =
105,506,000 J = 1.05506E15 erg = 25,199.67517 Kcal =
25,199,675.1696 cal = 100,000.393096 Btu = 77,817,232.1218 ftlb =
29.300111111 kWh 1 MeV = 0.000001602177 erg = 1,000,000 eV 1 eV =
1.602177eeE-19 J = 1.60217733E-12 erg = 0.000001 MeV 1 kWh =
3,600,000 J = 3.6 MJ = 2,655,220 ftlb 1 MMBtu = 1 dekatherm
POWER W = Watt Hp = horsepower J= Joule kW = kiloWatt btu/hr =
British Thermal Units per Hour
1 W = 1 s
J = 1
s
mN = 0.738
s
lbft = 3.41297
hrbtu
= 1.3410221E-3 hp = 10,000,000 s
erg
1 hp =745.69987 W = 550 s
lbft = 33,000
minlbft
= 2545.05076 hrbtu
= 0.746 kW
1 kW = 1000 W = 1.3410221 hp = 3412.96928 hrbtu
= 737.562 s
lbft
1 hrbtu
= 0.293 W
PRESSURE pa = Pascal atm = Atmosphere bar = Bar torr = Torr psi
= Pound per Square Inch inHg = Inches of Mercury inH20 = Inches of
Water mmHg = Millimeters of Mercury psf = Pound per Square Foot
1 pa = 12m
N= 9.86923E-6 atm = 0.00001 bar = 0.000145 psi = 0.00750062 torr
= 0.00750062 mmHg = 0.0002953 inHg = 0.00401865 inH20
1 atm = 101,325 pa = 1.01325 bar = 14.6959487755 psi = 760 torr
= 760 mmHg = 29.92126 inHg = 407.189359 inH20 1 bar = 100,000 pa =
0.9869233 atm = 14.503774 psi = 750.061683 torr = 750.061683 mmHg =
29.5299875 inHg = 401.864652 inH20 1 psi = 6894.757293 pa =
0.068046 atm = 0.0689476 bar = 51.714933 torr = 51.714933 mmHg =
2.036021 inHg = 27.7075924 inH20 = 144 psf 1 torr = 1 mmHg (same as
below) 1 mmHg = 133.32237 pa = 0.001315789 atm = 0.00133322 bar =
0.0193368 psi = 0.03937 inHg = 0.5357755 inH20 1 inHg = 3386.38816
pa = 0.033421053 atm = 0.03386388 bar = 0.491154077 psi = 25.4 torr
25.4 mmHg = 13.608697 inH20 1 inH20 = 248.84 pa = 0.00245586 atm =
0.0024884 bar = 0.0360912 psi = 1.8664535 torr = 1.8664535 mmHg =
0.073482421 inHg 1 Mpa = 1
2mm
N
1 kpa = 1 2m
kN
1 2cm
dyn = 0.1 pa
TEMPERATURE C = degrees Celsius F = degrees Fahrenheit K =
degrees Kelvin R = degrees Rankine
C = ( ) 15.27315.2739532
95
== KRF
F = 67.45967.4595932
59
==+ RKC
K = ( ) 15.2739515.27332
95
+==+ CRF
R = ( ) 67.4595915.273
59
+==+ FKC
RF = 67.4590 KC = 15.2730
-
ELECTRICITY V = Volt A = Ampere C = Coulomb = Ohm F = Farad W =
Watt Fdy = Faraday H = Henry mho = Mho S = Siemen T = Tesla Wb =
Weber
1 V = 1 2
2
sAmkg
= 1 A
1 A = 1 V
1 C = 0.0000103641 Fdy = 1 sA
1 = 1 32
2
sAmkg
= 1 AV
1 F = 1 2
42
mkgsA
1 W = 1 3
2
s
mkg = 1
s
J = 1
s
mN
1 Fdy = 96,487 C = 96,487 sA
1 H = 1 22
2
sAmkg
1 mho = 1 S = 1 2
32
mkgsA
1 S = 1 mho = 1 2
32
mkgsA
1 T = 1 2sA
kg
1 Wb = 1 2
2
sAmkg
ANGLES = Degree rad = Radian grad = Grade arcmin = Minute of Arc
arcs = Second of Arc
1 = 0.0174533 rad = 1.11111111 grad = 60 arcmin = 3600 arcs 1
rad = 57.29578 = 63.661977 grad = 3437.74677078 acrmin =
206,264.806247 arcs 1 arcmin = 1/60 = 1/3437.74677078 rad = 1/54
grad = 60 arcs 1 arcs = 1/3600 = 0.00000484814 rad = 1/3240 grad 1
sr = 1 22 mm
LIGHT fc = Foot Candle flam = Foot Lambert lx = Lux ph = Phot sb
= Stilb lm = Lumen cd = Candela lam = Lambert
1 fc = 10.763891 lx = 0.0010764 ph = 10.7639104167 2m
srcd
1 flam = 0.000342626 sb = 0.0010764 lam = 3.4262591 2m
cd
1 lx = 0.9290304 fc = 0.0001 ph = 12m
srcd
1 ph = 929.0304 fc = 10,000 lx = 1000 2m
srcd
1 sb = 2918.6350796 flam = 3.1415926 lam = 10,000 2m
cd
1 lm = 1 cdsr 1 cd = 1 candle power 1 lam = 929.0304 flam =
0.3183099 sb = 3183.09886184
2m
cd
RADIATION Gy = Gray rad = Rad rem = Rem Sv = Sievert Bq =
Becquerel Ci = Curie R = Roentgen
1 Gy = 100 rad = 100 rem = 1 Sv = 1 2
2
s
m
1 rad = 0.01 Gy = 1 rem = 0.01 Sv
-
1 rem = 0.01 Gy = 1 rad = 0.01 Sv 1 Sv = 1 Gy = 100 rad = 100
rem = 1
2
2
s
m
1 Bq = 2.7027E-11 Ci = 1 s
1
1 Ci = 37,000,000,000 Bq 1 R = 0.000258
kgsA
VISCOSITY P = Poise St = Stokes
1 P = 1 sm
kg
1 St = 0.0001 s
m2
FREQUENCY hz = Hertz cpm = Cycles per Minute cps = Cycles per
Second rev = Revolution rpm = Revolution per Minute
1 hz = 1 cps = 60 rpm
1 rpm = 1 cpm = 601 hz
1 rev = 360 deg 1000 rpm =
601000
hz
1 minrev
=
602pi
s
rad
Derivatives
Remember: dudxdu
u ==' , a = constant
( ) aaxdxd
=
[ ] 'auaudxd
=
[ ] 0=adxd
[ ] 1=xdxd
dxdu
dudv
dxdv
=
dudxdxdu
/1
=
[ ] '' vuuvuvdxd
+= (Chain Rule)
[ ] '' vuvudxd =
2''
v
uvvu
v
u
dxd
=
[ ] ( ) 'cossin uuudxd
=
[ ] ( ) 'sincos uuudxd
=
[ ] ( ) 'sectan 2 uuudxd
=
[ ] ( ) 'tansecsec uuuudxd
=
[ ] ( ) 'cotcsccsc uuuudxd
=
[ ] ( ) 'csccot 2 uuudxd
=
[ ] ( )111
'
sin2
1
= u
uu
uu
dxd
[ ] 21 1'
cotu
uu
dxd
+
=
( ) ( ) 'coshsinh uuudxd
=
( ) ( ) 'sinhcosh uuudxd
=
( ) 'loglog uu
eu
dxd
=
[ ]u
uu
dxd 'ln =
( ) ( ) 'ln uaaadxd uu
=
-
Integrals Remember: x is a variable, u is a variable, v is a
variable, a = constant, b = constant, C = constant, k = constant,
f(x) = function n = integer, fraction, and/or number m = integer,
fraction, and/or number
Caxadx +=
+= Cxdx ( ) ( ) += Cxfkdxxkf
( )11
1
++
=+
nCn
xdxxn
n
( )11
1
+
=
nCn
x
x
dx nn
+= Cxxdx ln
( ) += Cxxxdxx lnln ( ) += Cxaxxdxax lnln
( ) ( ) Cdxxfdxxfba
a
b+=
( ) ( ) ( )
+= bac
b
c
adxxfdxxfdxxf
Cedxe xx +=
Ca
edxeax
ax +=
( )
+=+ dvdudvdu
= vduuvudv (Integration by Parts)
Ca
adxau
u += ln
( ) ( )10ln
1>+= aaCa
a
bdxa
bxbx
Caxaxx
dx+= lnlnln
( ) ++=+ Cbxabbxadx ln1
( ) ++= Caxxax dx 2222 ln
+=
Cxaxa
xdx 2222
+=Cax
ax
xdx 2222
Ca
x
aaxx
dx+=
1
22sec
1
Cxxx
dx+=
1
2sec
1
( ) ( ) ++=+
Cm
axdxaxxm
mn
1lnln
1
( )1m ( ) Cax
a
edxxeax
ax += 12
+= Cdxex
a
nex
adxex axnaxnaxn 11
( ) += Caxaedxxe
axax 12
= babxdxbx
axnaxn
ln
( ) ( )10ln
1 >+ bbCbxdxx
ban n
+=+ Cx
x
dx 12 tan1
Cx
a
axa
dx+=
+1
22 tan1
Cxa
xa
axa
dx+
+=
ln21
22
( )022 > xa C
ax
ax
aax
dx+
+
=
ln21
22
( )022 > ax ( ) Cxa
axa
xdx+=
22
22 ln21
( ) Cxax
aaxx
dx+
+=
+ln1
( ) ( )( ) ( )111
++
+=+
+
nCnbbxadxbxa
nn
( ) ( ) ++=+ Cbxabbxadx 1
2
( ) ( ) ++==+ Cbxabbxadx
23 21
=+ 222 xbadx
Ca
xba
bxb
+
++ 2
22
1ln1
= dxxba222
Ca
bxb
axbax +++ 1
2222 sin
22
+=
Ca
bxbxba
dx 1222
sin1
( ) Cbeaaca
x
beadx cx
cx++=
+ln1
( ) += Cxdux cossin ( ) += Cxdux sincos ( ) += Cxdxx coslntan (
) ++= Cxxdxx cotcsclncsc ( ) Cxxdxx ++= tanseclnsec ( ) += Cxdxx
sinlncot ( ) += Caxadxax cos
1sin
( ) += Caaxdxax sincos
( ) ( ) =+= Caxadxax cosln1
tan
( ) Caxa
+secln1
( ) ( ) =+= Caxaxadxax cotcscln1
csc
Caxa
+
2tan
1
( ) ( ) =++= Caxaxadxax tansecln1
sec
Caxa
+
+
42tan
1 pi
( ) ( ) Caxa
dxax += sinln1
cot
( ) Cxxdxx += 42sin
2sin 2
( ) Cxxdxx ++= 42sin
2cos2
( ) += Cxxdxx tantan 2 ( ) += Cxdxx tansec2 ( ) Cxdxx += cotcsc2
( ) += Cxdxx sinlncot 2 ( ) ( ) Cbxabbxa
dx+
+=
+ 34 31
( )[ ] ++=+ Cbxaabxbbxaxdx ln12
( ) ( ) +
++
+=
+Cbxa
bxaa
bbxaxdx ln122
( ) ( )( )( ) ++=+ Cbxabnbxadx
nn 111
,
( )1n ( ) ++=+ Cbxabdxbxa
3
32
-
( ) Cxxx
xd+=
1
2csc
1
=
a
bxabxba
dx 1222 tan
1 (x in rads)
( )0,0 >> ba =+=
C
a
bxabxba
dx 1222 tanh
1
Cbxabxa
ab+
+ln2
1 (x in rads)
( ) ( ) ++
=
+ 223 22
bxabbxa
bxaxdx
( ) ( ) ++
=
+ 324 63
bxabbxa
bxaxdx
=
+ 32
21bbxa
dxx
( )( ) ( )[ ] Cbxaabxabxa +++++ ln23 2
( ) =+ 22
bxadxx
( ) ( ) Cbxaabxa
bxabxb
+
+
+
+ ln2213
( ) =+ 32
bxadxx
( )( ) ( ) Cbxabxa
bxaab
+
++
+
+ ln2
43123
( ) ( ) ++++
=
+C
bxabxbabxa
bxadxx
33
222
4
2
333
++=+222222
2xbaxdxxba
Ca
xba
bxb
a+
++ 2
222
1ln2
=+ bxaxdx
2
Cbxax
dxa
bax
bxa+
++
+ 2
( ) =+ 222 xadx
( ) Cax
axaa
x++
+1
3222 tan21
2
( ) = 222 xadx
( ) Cxadx
axaa
x+
+
222222 21
2
( ) ++=+ Caxax
ax
dx2222/322
( ) ++=+ Caxaxxdx
222/322
1
( ) = 322222 4 axxdxaxx
( ) Caxxaaxxa ++ 224222 ln88
( ) += 322222 4 xaxdxxax
Ca
xaxax
a+
+ 122
2
sin8
(a >0)
=+ dxbxax
( ) ( )C
bbxabxa
++
2
3
15322
=+ dxbxax2
( ) ( )C
bbxaxbabxa
+++
3
3222
105151282
( ) ++=+ Cabxbbxaxdx 2
2 ln21
?tan 12
2
+=+
Cabxabb
a
bx
bxadxx
=++ 2cxbxa
dx
Cc
bcxcxbxa
c+
++++
2ln1 2
( )04
2sin1
2
1 >+
= cC
acbbcx
c
( ) Cbxaba
bx
bxaxdx
++=+
ln2
( ) ( )+
= 122
22
n
axxdxaxn
( ) ( )11
222
2
++
nCdxaxn
na n
( ) = naxdx
22
( )( ) ( ) +
Cax
dxan
axxn 2
222
22
2
( )2n ( ) ( )
2
222
22 ++
=+
Cn
axdxaxxn
n
= dxxa22
Ca
xaxax +
+ 1222 sin
21
( ) += Cxadxxax 2/32222 31
= dxax22
( )[ ]Caxxaaxx ++ 22222 ln21
( ) Caxdxaxx += 2/32222 31
=+=
Ca
x
xa
dx 122
sin
( )0cos 221 >+ xaCa
x
=+=+
Cxa
xa
dx 122
sin
( ) Cxax +++ 22ln
=+=
Ca
x
ax
dx 122
cosh
Caxx ++ 22ln
=
axdx
x
xa 12
22
sin
Cx
xa+
22
( )++=+ 222 22 ln xaxdxx xa C
x
xa+
+ 22
Cxa
ax
axx
dx+
=
222
222
=
222 xaxdx
Cx
xaa
a+
++
22
ln1
=
22 xaxxdx
Cxaxa
axa +
21 2sin
-
( ) ++=+ 2222222 28 xaxaxdxxax
( ) Cxaxa +++ 224 ln8
=+ dxx
xa 22
Cx
xaaaxa +
+++
2222 ln
=+ 22
2
xa
x
( ) Cxaxxaxa +++++2
ln2
2222
2
++
=
+C
xa
xa
xax
dx2
22
222
==
a
xadxxax 14
222 sin8
( ) Cxaxax + 2222 281
= dxx
xa 22
Cx
xaaxa +
+
2222 ln
+
=
Cxa
xadxxa
x2
22
22
2
= dxax22
Caxxaaxx +++ 222
22 ln22
( ) Ca
axxdxax += 42sin
2sin 2
( ) ++= Caaxxdxax
42sin
2cos2
( ) ( ) Cxaxa
dxax += tan1
tan 2
( ) += Caxadxax cot1
csc2
( ) += Caxadxax tan1
sec2
( ) =+= Cxaxadxax cot1
cot 2
Caxa
+ sinln1
= dxx
ax 22
Cx
aaax + 122 sec
=
dxxax
2
22
Cx
axaxx +
+22
22ln
=
dxax
x
22
2
Caxxaxxa +++ 22222
2ln
2
=
22 axx
dx
Cx
a
aC
a
x
a+=+ 11 cos
1sec
1
Ca
ax
xax
dx+
=
1
2sin
2
=
22 xaxdx
Ca
axaxax
ax+
+
12
2 sin2
22
( ) ( )( ) + = 1222
2
n
xaxaxdxxaxn
n
( ) +++ Cdxxaxnna n 222
21
( )( )( )
( ) +
2
22
2 22
2 anxaxax
xax
dxa
n
( )( ) ( ) +
Cxax
dxan
nn 2
22
223
( ) = dxax1cos Cxa
aaxx + 221 11cos
( ) = dxax1tan ( ) Cxaaxx ++ 221 1ln
21
tan
( ) Ca
axx
a
axdxaxx +=cossin
sin 2
(x in rads) ( ) C
a
axx
a
axdxaxx ++=sincos
cos 2
(x in rads)
( ) =+ dxbxax n
( ) Cn
a
n
bxabbxa n
+
+
+
++ +
1221
,
( )2,1 n ( ) ( ) C
n
bxab
dxbxan
n
++
+=+
+
22
2
,
( )2n
=
+ dxx
bxa
++++ C
bxaxdx
abxa2
tan2 1 +
+
=
+ C
a
bxaabxax
dx
ln1 +++
+=
+C
abxaabxa
abxaxdx
=+ dx
x
bxa2
Cbxax
dxbx
bxa+
++
+ 2
+
=
aaxadxxaxx 1
32 sin
22
( )( ) Cxaxaxax ++6
232 2
=
dxxxax 22
Ca
axaxax +
+ 12 sin2
=
dxxxax
2
22
Ca
ax
x
xa+
1sin22
Cx
xa
axaxx
dx+
=
21
2 2
( ) ( )( ) ++
=
baxbadxbxax
2cos
cossin
( )( ) ( )222
cos baCba
xba+
( ) ( )( )
= baxbadxbxax
2sin
sinsin
( )( ) ( )222
sin baCba
xba+
+
+
-
( ) ( ) +=
na
axaxdxaxn
n cossinsin1
( ) Cdxaxn
n n +
2sin1
( ) +=
na
axaxdxaxn
n sincoscos1
Caxn
n n +
2cos
1
( ) =dxaxntan ( ) ( ) ( )1tan1
tan 21
+
nCdxaxna
ax nn
( ) ( )( ) +=
1cotcsc
csc2
na
axaxdxaxn
n
( ) ( )1csc12 2 +
nCdxaxn
n n
( ) ( )( ) +=
1tansec
sec2
na
axaxdxaxn
n
( ) ( )1sec12 2 +
nCdxaxn
n n
( ) ( ) =
1cot
cot1
na
axdxaxn
n
( ) ( )1cot 2 + nCdxaxn ( ) = dxax1sin
Cxaa
axx ++ 221 11sin
Caxaax
dx+= cot
1sin 2
Caxaax
dx+= tan
1cos2
+
+=
Caxaax
dx24
tan1
sin1pi
(x in rads) Cax
aax
dx+
=
+ 24tan
1sin1
pi
(x in rads)
+=
Caxaax
dx2
cot1
cos1
+=+Cax
aax
dx2
tan1
cos1
=+ axcbdxsin
+
24tantan
2 122
ax
cbcb
cbapi
( )22 cbC >+
( ) =dxaxx sin2 Cax
a
xaax
a
x+
+ cos
2sin2 3
22
2
( ) = dxaxx cos2 Cax
a
xaax
a
x+
+ sin2cos2 322
2
( ) =dxaxx n sin ( ) Cdxaxx
a
nax
a
x nn
++ coscos 1
( ) = dxaxx n cos ( ) Cdxaxx
a
nax
a
x n + sinsin 1
2
( ) = dxaxx n 1sin C
xa
dxxn
aax
n
x nn+
+
+ +
+
22
11
1
11sin
1
( )1n ( ) ++=
+ ax
n
xdxaxxn
n 11
1 cos1
cos
( )111 22
1
+
+ +
nCxa
dxxn
a n
( ) +=
axn
xdxaxxn
n 11
1 tan1
tan
( )111 22
1
++
+
nCxa
dxxn
a n
( ) += Cxdxxx csccotcsc ( ) += Cxdxxx sectansec ( ) += Ca
axdxaxax4
2coscossin
( ) += Caxadxax sinh1
cosh
( ) ( ) += Caxadxax coshln1
tanh
( ) += Caxadxhax 2tanhln1
csc
( ) ( ) += Caxadxhax tanhsin1
sec 1
( ) += Caxadxax sinhln1
coth
( ) += Cxaaxdxax
242sinh
sinh 2
( ) ++= Cxaaxdxax
242sinh
cosh 2
( ) += Caxaxdxax tanh1
tanh 2
( ) ( )( ) +
=
baxbadxbxax
2sin
coscos
( )( ) ( )222
sin baCba
xba+
+
+
( ) ( ) ++=+
Can
axdxaxaxn
n
1sin
cossin1
( )1n ( ) ( ) ++=
+
Can
axdxaxaxn
n
1cos
sincos1
( )1n ( ) += Cna
axdxaxaxn
n csccotcsc
( )0n ( ) C
na
axdxaxaxn
n +=sec
tansec
( )0n ( ) = dxaxax mn cossin
( )( ) ++
+
nma
axax mn 11 cossin
( ) ++
Cdxaxaxnm
n mn cossin1 2
( )mn ( ) = dxaxax mn cossin
( )( ) ++
+
nma
axax mn 11 cossin
( ) +++
Cdxaxaxnm
m mn 2cossin1
( )nm ( ) =dxbxeax sin
( ) Cbxbbxaba
eax+
+cossin22
( ) =dxbxeax cos ( ) Cbxbbxa
baeax
++
sincos22
( ) =dxaxx sinh Cax
aax
x
a+ sinh1cosh 2
( ) =dxaxx cosh Cax
aax
x
a+ cosh1sinh 2
-
=+ axcbdxsin
axcbaxbcaxbc
bca sincossinln1
22
22 +
++
( )22 bcC >+
=+ axcb
dxcos
Caxcbcb
cba+
+
2tantan
2 122
( )22 cb >
=+ axcb
dxcos
Caxcb
axbcaxbcbca
++
++
cos
sincosln122
22
( )22 cb < ( ) ( ) += Cxdxx coshsinh ( ) ( ) += Cxdxx
sinhcosh ( ) ( ) += Cxdxx coshlntanh ( ) += Cxdxhx 2tanhlncsc ( ) (
) += Cxdxhx tanhsinsec 1 ( ) += Cxdxx sinhlncoth ( ) += Caxadxax
cosh
1sinh
( ) += Caxadxaxh coth1
csc 2
( ) += Caxadxaxh tanh1
sec 2
( ) += Caxaxdxax coth1
coth 2
( ) ( ) =
na
axaxdxaxn
n cossinhsinh1
( ) ( )0sinh1 2 + nCdxaxnn n
( ) ( ) +=
na
axaxdxaxn
n sincoshcosh1
( ) ( )0cosh1 2 + nCdxaxnn n
( ) Cdxaxn += tanh ( ) ( ) Cdxaxan
ax nn
++
21
tanh1
tanh
( )1n ( ) ( ) +=
an
axaxhdxaxhn
n
1cothcsc
csc2
( ) ( )1csc12 2 +
nCdxaxhn
n n
( ) ( ) +=
an
axaxhdxaxhn
n
1tanhsec
sec2
( ) ( )1sec12 2 +
nCdxaxhn
n n
( ) ( ) +=
an
axdxaxn
n
1coth
coth1
( ) ( )1coth 2 + nCdxaxn
( ) =dxaxx n sinh ( ) Cdxaxx
a
nax
a
x nn
+ coshcosh 1
( ) =dxaxx n cosh ( ) Cdxaxx
a
nax
a
x nn
+ sinhsinh 1
( ) Cna
axhdxaxaxhn
n +=sec
tanhsec
( )0n ( ) C
na
axhdxaxaxhn
n +=csc
cothcsc
( )0n ( ) =dxbxe ax sinh
( )222
baCba
e
baee bxbxax
+
+
( ) =dxbxe ax cosh
( )222
baCba
e
baee bxbxax
+
++
