Definition of the Six Trigonometric Functions Right triangle definitions, where 0 < < / 2.

..

tan

..

cos

.

.sin

adjopphypadjhypopp

=

=

=

θ

θ

θ

..

cot

..

sec

.

.csc

oppadjadjhypopphyp

=

=

=

θ

θ

θ

Circular Function Definitions, where is any angle

xyrxry

=

=

=

θ

θ

θ

tan

cos

sin

yxxryr

=

=

=

θ

θ

θ

cot

sec

csc

Units Circle

707.02

122 == 866.0

23 = 500.0

21 =

Common Reference Triangles

Reciprocal Identities

xx

xx

tan1

cot

cot1

tan

=

=

xx

xx

sec1

cos

csc1

sin

=

=

xx

xx

cos1

sec

sin1

csc

=

=

Tangent and Cotangent Identities

xx

xcossin

tan = xx

xsincos

cot =

Hyperbolic Functions

2sinh

xx eex

−−= 2

coshxx ee

x−+=

xx

xcoshsinh

tanh =

xx eexhx −−

== 2sinh

1csc

xx eexhx −+

== 2cosh

1sec

xx

xx

sinhcosh

tanh1

coth ==

Sum and Difference Formulas ( )( ) vuvuvu

vuvuvu

sinsincoscoscossincoscossinsin

=±=±

( )vuvu

vutantan1tantan

tan

±=±

Double-Angle Formulas

uuu cossin22sin = u

uu 2tan1

tan22tan

−=

uuuuu 2222 sin211cos2sincos2cos −=−=−= Power Reducing Formulas

( )uu

u 2cos121

2cos21

sin 2 −=−= uu

u2cos12cos1

tan 2

+−=

( )uu

u 2cos121

2cos21

cos2 +=+=

xx

xx

x2cos1

2sin2sin

2cos1tan

+=−=

Sum-to-Product Formulas

−

+−=−

−

+=+

−

+=−

−

+=+

2sin

2sin2coscos

2cos

2cos2coscos

2sin

2cos2sinsin

2cos

2sin2sinsin

vuvuvu

vuvuvu

vuvuvu

vuvuvu

Reduction Formulas ( )( ) xx

xx

csccscsinsin

−=−−=−

( )( ) xx

xx

tantancoscos

−=−−=−

( )( ) xx

xx

cotcotsecsec

−=−−=−

Pythagorean Identities

1cossin 22 =+ xx xx 22 sectan1 =+ xx 22 csccot1 =+

Fractions

bdac

dc

ba

bdbcad

dc

ba

=⋅

+=+

bcad

cd

ba

dcba

=⋅= b

aba

ba

−=−=−

( ) ( )( ) ( )

( ) ( )( ) ( )

( )( )bdfgeh

fhbcadbdfhbdfh

hgfedcba

hgfedcba

++=⋅

++=

++

////

////

Exponents

mm

nmn

m

mnmn

AA

AAA

AAA

1=

=

=

−

−

+

( )( )

n

nn

nnn

nmmn

BA

BA

BAAB

AA

=

=

=

( )

nn

mnn mnm

AA

AA

AAA

=

=

==

1

2/1

/

( )010 ≠= AA n

nn

B

ABA = nnn BAAB =

Logarithms log = common logarithm (logarithm to the base 10) = 10log

ln = natural logarithm (logarithm to the base e) = elog e = 2.71828182846

Ae

BA

A

A

B

==

=

ln

log

10

A

BA

Ae

A

B

==

=

10log

ln

01ln1log10 log

ln

===

=

A

AeA

A

xBA

e

==

=1ln

110log

BABA

BAAB

logloglog

logloglog

−=

+=

AnA

AA

n loglog

log1

log

=

−=

BA

xloglog=

BU

UA

AB log

loglog = xx 10loglog = xx elogln =

( )( ) AAA log30258509.2log10lnln ==

( )( ) ( ) AAAeA ln43429448.0lnlnlnloglog === Law of Sines Law of Cosines

Cc

Bb

Aa

sinsinsin== Cabbac cos2222 −+=

CabBacAbcArea sin21

sin21

sin21 ===

°=++ 180CBA

Similar Triangles

cc

bb

aa ''' ==

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

=

−

=

−

=

−

π

π

π

xx

xx

xx

tan2

cot

cot2

tan

sin2

cos

=

−

=

−

=

−

π

π

π

xx

xx

xx

cos2

sin

sin2

cos

cos2

sin

=

+

=

−

−=

−

π

π

π

2cos1

2sin

2cos1

2cos

sin2

cos

xx

xx

xx

−±=

+±=

−=

+ π

xxx

sincos1

2tan

−=

Quadradic Equation and Quadradic Formula

)0(02

≠=++

a

cbxax

aacbb

x2

42 −±−=

Temperature Conversion Formulas

( )

( )

( ) 67.45959

15.27359

15.27395

15.2733295

67.45967.45959

3259

15.27315.27395

3295

15.273067.4590

+°=°=+°=°

+°=°=+−°=°

−°=−°=+°=°

−°=−°=−°=°

°=°°=°

FKCR

CRFK

RKCF

KRFC

KCRF

Distance Between Points ( )11 , yxP and ( )22 , yxQ is,

( ) ( ) ( )221

221, yyxxQPd ++−=

Slope of Line Between Points 12

12

xxyy

m−−

=

Distance Between any Point ( )11 , yxP and any line 0=++ CBxAx is,

22

11

BA

CByAxd

+

++=

Factoring Formulas ( )( )

( ) 222

22

2 yxyxyx

yxyxyx

+±=±

−+=− ( )( )( ) 32233

2233

33 yxyyxxyx

yxyxyxyx

±+±=±

+±=±

The Greek Alphabet Alpha Beta Gamma Delta Epsilon Zeta Eta Theta θ

ηζεδγβα

ΘΗΖΕ∆ΓΒΑ

Iota Kappa Lambda Mu Nu Xi Omicron Pi π

οξνµλκ

ι

ΠΟΞΝΜΛΚΙ

Rho Sigma Tau Upsilon Phi Chi Psi Omega ϖ

ψχφυτσρ

ΩΨΧΦΥΤΣΡ

Binomial Theorem For any positive integer n,

( ) ( )

( )( ) nnn

nnnn

bnabbannn

bann

bnaaba

+++⋅⋅

−−+

⋅−++=+

−−

−−

133

221

...321

2121

1

For instance,

( )( )( )( ) 4322344

32233

222

1

464

33

2

babbabaaba

babbaaba

bababa

baba

++++=+

+++=+

++=+

+=+

Difference of Like Integers Powers, n > 1 ( )( )122321 ... −−−−− +++++−=− nnnnnnn babbabaababa

For instance,

( )( )( )( )( )( )322344

2233

22

babbaababa

babababa

bababa

+++−=−++−=−

+−=−

Series Formulas

...!7!5!3

sin753

+−+−= xxxxx ...

!6!4!21cos

642

+−+−= xxxx

...!7!5!3

sinh753

++++= xxxxx ...

!6!4!21cosh

642

++++= xxxx

...!4!3!2

1432

+++++= xxxxe x

( )11...,753

tan753

1 ≤≤−+−+−=− xxxx

xx

( )11...,11

1 432 <<−+++++=−

xxxxxx

( ) ( )11...,432

1ln432

≤<−+−+−=+ xxxx

xx

Simple Interest Compound Interest

tI Pr= nt

nr

PA

+= 1

I = Interest, P = Principle, r = Rate, t = Time of Investment A = Money Accrued, n = Number of Periods in Time

Degree Conversions π = 3.14159265359

π2 radians = 360 degrees 2π

radians = 90 degrees

1 radian = π

180 degrees = 57.2957795131º

1 degree = 180π

radians = 0.01745329252º

To get radians from degrees, multiply the degrees by 180π

To get degrees from radians, multiply the radians by π

180

Interpolation Equation (b is unknown)

fed

cba ?

efbc

deab

−−=

−−

( ) ( )

( )dfdecefa

b−

−+−=?

Example:

211496?2

1421?6

9142?

−−=

−−

( ) ( )

( ) 667.3921

914614212? =

−−+−=

GPS Coordinates 1 Degree = 1° = 60’ = 3600” = 60 miles = 316800 feet 1 Minute = 1’ = 60” = 1 mile = 5280 feet 1 Second = 1” = 88 feet Convert Degrees Minutes Seconds to Decimal Degrees DDD° MM’ SS.sssssss” DDD.ddddddd° (i.e. 109° 40’ 04.04832” 109.6677912°)

6060

.

.

sssssssSSMM

DDDdddddddDDD+

+=°

Convert Degrees Minutes Seconds to Degrees Decimal Minutes DDD° MM’ SS.sssssss” DDD° MM.mmmmmmm’ (i.e. 109.6677912° 109° 40.0647200’)

++=°60

.'.

sssssssSSMMDDDmmmmmmmMMDDD

Convert Decimal Degrees to Degrees Minutes Decimal Seconds DDD.ddddddd° DDD° MM’ SS.sssssss”

( )( )

( )( ) 6060."..060.'

60.'..

⋅−⋅−=−⋅−=

⋅−=−=°

MMDDDdddddddDDDsssssssSS

mmmmmmmDDDdddddddDDDMM

DDDdddddddDDDmmmmmmmMM

DDDdddddddDDDDDD

Convert Decimal Degrees to Degrees Decimal Minutes DDD.ddddddd° DDD° MMM.mmmmmmm’

( )DDDdddddddDDDmmmmmmmMM

DDDdddddddDDDDDD

−=−=°

.'..

Triangle

bhArea

ah

21

sin

=

= θ

Law of Cosines θcos2222 abbac −+=

Right Triangle Pythagorean Theorem

222 bac +=

Equilateral Triangle

243

23

2 shsArea

sh

==

=

32h

s =

Parallelogram

bhArea =

Trapezoid

( )bah

Area +=2

Circle

rDnceCircumfere

DDrArea

DDr

rD

ππ

πππ

242

2212

222

==

=

==

==

=

Sphere

2

3

434

rAreaSurface

rVolume

π

π

=

=

Circular Ring ( p = average radius, w = width of ring)

( )22 rRArea −= π pwπ2=

Sector of a Circle (θ in Radians)

22

2

2

2

==

Dr

Areaθ

θ

28

2 rsD == θ

2D

rsθθ ==

Sector of a Circular Ring ( p = average radius, w = width of ring,θ in Radians)

pwArea θ=

Ellipse

abcVolumeEllipsoid

banceCircumfere

abArea

π

π

π

43

22

22

=

+≈

=

Right Circular Cone

3

2hrVolume

π=

Lateral Surface Area =

rshrr ππ =+ 22

Any Cone or Pyramid (A = Area of Base)

3Ah

Volume =

Right Circular Cylinder

hrVolume 2π= Lateral Surface Area = rhπ2

Wedge (A = area of upper face, B = area of base)

θsecBA =

Frustum of Right Circular Cone

( )3

22 hRrRrVolume

++= π

Lateral Surface Area =

( )rRs +π

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

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

−

−

−

−

−

−

−

−

−

−

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 µ

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 Avogadro’s 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

B me

2=µ 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

Boltzmann’s Constant A

B NR

k = 1.380 658(12) x 2310− J/K

7.270 063 x 2710− Btu/Rº

Compton Wavelength cm

h

eC =λ 2.426 310 58(22) x 1210− m

Density of Air (20ºC and 1 atm) 1.20 kg/ 3m 0.074 913 600 lbm/ 3ft Density of Water (20ºC 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

Dirac’s Constant π2h= 1.054 572 66(63) x 3410− J·s

2.77 505 62 x 4110− Btu·hr 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 Faraday’s Constant F 964785.309 C/gmol Gas Constant R 8.314 510(70) J/K·mol 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 =Φ 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 p

n me

2=µ 5.050 786 6(17) x 2710− J/T

Permeability of Free Space 0µ 4π x 710− N/ 2A (exact)

Permittivity of Free Space 20

0

1cµ

ε = 8.854 187 817 x 2710− 2C /N· 2m (exact)

Photon Wavelength 0λ 1239.8425 nm

12398.425

A Photon Frequency 0f 2.417 988 3 x 1410 Hz

Planck’s Constant h 6.626 075(40) x 3410− J·s 1.744 529 9 x 4010− Btu·hr 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 2eh

25812.805 6(12) Ω

Rydberg Constant HR 1.097 373 153 4(13) x 710 1−m 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 σ 5.670 510 x 810 W/ 42 Km ⋅ 1.712 340 x 910− Btu/h· 2ft ·º 4R Wein’s Constant 3c 0.002 897 756 m·K 5215.961 µ · ºR 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 µ = 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 µ = 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 µ = 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 µ = 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 µ = 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 µ = 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 µ = 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 µ = 254,000 A = 25,400,000,000 fermi 1 µ = 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 µ = 100,000 fermi 1 fermi = 1E-15 m = 1E-13 cm = 1E-12 mm = 3.937E-14 in = 3.937E-11 mil = 0.000000001 µ = 0.00001 A AREA m² = Meters Squared cm² = Centimeters Squared B = Barn yd² = Yard squared ft² = Feet squared in² = Inches Squared km² = Kilometers Squared Ha = Hectar A = Are mi² = Miles squared Acre = Acre miUS² = US Statute Miles Squared 1 m² = 10,000 cm² = 1.19599 yd² = 10.7639 ft² = 1550.0031 in² = 3.86102E-7 mi² = 2.471044E-4 Acre = 1E28 B 1 cm² = 0.155 in² = 1.0764E-3 ft² 1 yd² = 0.83613 m² = 9 ft² = 1296 in² 1 ft² = 0.11111 yd² = 144 in² = 929.0304 cm² = 0.092903 m² 1 in² = 6.4516 cm² = 6.9444E-3 ft² = 0.00064516 m² 1 km² = 100 Ha = 10,000 A = 247.1044 Acre = 0.386102 mi² 1 Ha = 0.01 km² = 100 A = 2.471044 Acre = 3.86102E-3 mi² 1 A = 2.471044E-4 Acre = 0.01 Ha = 0.0001 km² = 3.86102E-5 mi² = 1E30 B 1 mi² = 2.589988 km² = 639.99744 Acre = 0.999996 miUS² = 3,097,600 yd² = 27,878,400 ft² = 2,589,933.11034 m² = 258.998811 Ha 1 miUS² = 1.000004 mi² 1 Acre = 4046.8726 m² = 4840.01936 yd² = 43,560.17424 ft² = 2.5625E-3 mi² = 4.046873E-3 km² = 0.4046873 Ha = 40.46873 A 1 B = 1E-28 m² VOLUME m³ = Meters Cubed st = Stere cm³ = Centimeters Cubed yd³ = Yards Cubed \ft³ = Feet Cubed in³ = 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 m³ = 1 st = 1,000,000 cm³ = 1.30795 yd³ = 35.314667 ft³ = 61023.7440946 in³ = 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 m³ (same as above) 1 cm³ = 0.000001 m³ = 0.000035315 ft³ = 0.061024 in³ = 1000 m³ = 0.000264172 gal = 0.0010567 qt = 0.0021134 pt = 1 ml = 0.004226753 cu

= 0.033814 oz = 0.067628 tbsp = 0.202884 tsp 1 yd³ = 0.76455486 m³ = 764,554.857984 cm³ = 27 ft³ = 46,656 in³ = 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 ft³ = 0.02832 m³ = 28,316.8466 cm³ = 0.037037 yd³ = 1728 in³ = 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 in³ = 16.387064 cm³ = 0.000578704 ft³ = 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 m³ = 1000 cm³ = 61.023744 in³ = 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 m³ = 3,785.411784 cm³ = 0.13368055 ft³ = 231 in³ = 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 m³ = 946.352946 cm³ = 57.75 in³ = 0.946352946 L = 946.352946 cm³

= 0.02685522 bu = 0.107451 pk 1 pt = 0.125 gal = 0.5 qt = 2 cu = 16 oz = 32 tbsp = 96 tsp = 473.176473 cm³ = 28.875 in³ = 0.473176 L = 473.176473 ml = 0.0134276 bu = 0.05371 pk 1 ml = 1 cm³ (see 1 cm³ above) 1 cu = 8 oz = 16 tbsp = 48 tsp = 0.25 qt = 0.5 pt = 0.0625 gal = 236.5882365 cm³ = 236.5882365 ml = 14.4375 in³ = 0.23658824 L

1 oz = 1.0408423 ozUK = 2 tbsp = 6 tsp = 29.57353 ml = 29.57353 cm³ = 0.029574 L = 0.0078125 gal = 0.03125 qt = 0.0625 pt = 1.8046875 in³ 1 ozUK = 0.96076036 oz 1 tbsp = 3 tsp = 0.5 oz = 0.0625 cu = 14.7867648 ml = 14.7867648 cm³ = 0.0147868 L = 0.00390625 gal = 0.015625 qt = 0.03125 pt = 0.90234375 in³ 1 tsp = 0.333333 tbsp = 0.1666667 oz = 0.020833334 cu = 4.9289216 ml = 4.9289216 cm³ = 0.00492892 L = 0.00130208 gal = 0.0104167 pt = 0.3008 in³ 1 bbl = 4.51168 bu = 18.0467072 pk = 67.375 fbm = 0.1589873 m³ = 5.61458334 ft³ = 0.20794753 yd³ = 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 m³ = 1.244456 ft³ = 0.046091 yd³ = 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 m³ = 0.311114 ft³ = 0.0115227 yd³ = 2.3272944 gal = 8.8097675 L = 9.3091774 qt 1 fbm = 0.2678546 pk = 0.066964 bu = 0.0148423 bbl = 0.00235074 m³ = 0.083334 ft³ = 0.00308642 yd³ = 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/s² = Meters per Second Squared ft/s² = Feet per Second Squared cm/s² = Centimeters per Second Squared 1 m/s² = 3.280835 ft/s² = 100 cm/s² 1 ft/s² = 0.3048 m/s² = 30.48 cm/s² 1 cm/s² = 0.01 m/s² 9.80665 m/s² = 32.174 ft/s² 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 = 12smkg ⋅ = 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 ft·lb = Foot – Pound therm = Therm (EEC) MeV = Mega electron Volt eV = electron Volt kWh = kiloWatt hour

1 J = 1 N·m = 1 W·s = 1 2

2

smkg ⋅ = 10,000,000 erg = 0.0002388 Kcal = 0.238846 cal = 0.00094782 Btu = 0.737562 ft·lb = 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 ft·lb = 0.00003968 therm 1 cal = 4.1868 J = 41,868,000 erg = 0.0.001 Kcal = 0.00396832 Btu = 3.0880252 ft·lb = 3.9683E-8 therm 1 Btu = 1055.05585 J = 0.251996 Kcal = 251.99576 cal = 778.16926 ft·lb = 0.000010002 therm = 0.000293071 kWh 1 ft·lb = 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 ft·lb = 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 ft·lb 1 MMBtu = 1 dekatherm POWER W = Watt Hp = horsepower J= Joule kW = kiloWatt btu/hr = British Thermal Units per Hour

1 W = 1 sJ = 1

smN ⋅ = 0.738

slbft ⋅

= 3.41297 hrbtu

= 1.3410221E-3 hp = 10,000,000 s

erg

1 hp =745.69987 W = 550 slbft ⋅

= 33,000 min

lbft ⋅ = 2545.05076

hrbtu

= 0.746 kW

1 kW = 1000 W = 1.3410221 hp = 3412.96928 hrbtu

= 737.562 slbft ⋅

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 ºC = degrees Celsius ºF = degrees Fahrenheit ºK = degrees Kelvin ºR = degrees Rankine

ºC = ( ) 15.27315.27395

3295 −°=−°=−° KRF

ºF = 67.45967.45959

3259 −°=−°=+° RKC

ºK = ( ) 15.27395

15.2733295 +°=°=+−° CRF

ºR = ( ) 67.45959

15.27359 +°=°=+° 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 Ω⋅A

1 A = 1 ΩV

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

smkg ⋅ = 1

sJ = 1

smN ⋅

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

2msrcd ⋅

1 flam = 0.000342626 sb = 0.0010764 lam = 3.4262591 2m

cd

1 lx = 0.9290304 fc = 0.0001 ph = 12msrcd ⋅

1 ph = 929.0304 fc = 10,000 lx = 1000 2msrcd ⋅

1 sb = 2918.6350796 flam = 3.1415926 lam = 10,000 2m

cd

1 lm = 1 cd·sr 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

sm

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

sm

1 Bq = 2.7027E-11 Ci = 1 s1

1 Ci = 37,000,000,000 Bq

1 R = 0.000258 kg

sA ⋅

VISCOSITY P = Poise St = Stokes 1 P = 1

smkg

⋅

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 = 60

1000 hz

1 minrev =

602π

srad

Derivatives

Remember: dudxdu

u ==' , a = constant

( ) aaxdxd =

[ ] 'auaudxd =

[ ] 0=adxd

[ ] 1=xdxd

dxdu

dudv

dxdv =

dudxdxdu

/1=

[ ] '' vuuvuvdxd += (Chain Rule)

[ ] '' vuvudxd ±=±

2

''v

uvvuvu

dxd −=

[ ] ( ) 'cossin uuudxd =

[ ] ( ) 'sincos uuudxd −=

[ ] ( ) 'sectan 2 uuudxd =

[ ] ( ) 'tansecsec uuuudxd =

[ ] ( ) 'cotcsccsc uuuudxd −=

[ ] ( ) 'csccot 2 uuudxd −=

[ ] ( )111

'sin

2

1 <<−−

=− uu

uu

dxd

[ ] ( )111

'cos

2

1 <<−−

−=− uu

uu

dxd

[ ]2

1

1'

tanu

uu

dxd

+=−

[ ] ( )11

'sec

2

1 >−

=− uuu

uu

dxd

[ ] ( )11

'csc

2

1 >−

−=− uuu

uu

dxd

[ ]2

1

1'

cotuu

udxd

+−=−

( ) ( ) 'coshsinh uuudxd =

( ) ( ) 'sinhcosh uuudxd =

( ) 'log

log uu

eu

dxd

=

[ ]uu

udxd '

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

−≠++

= +

nCnx

dxxn

n

( )11

1

≠+−

= −

nCn

xxdx n

n

+= Cxx

dxln

( ) +−= Cxxxdxx lnln

( ) +−= Cxaxxdxax lnln

( ) ( ) Cdxxfdxxfb

a

a

b+−=

( ) ( ) ( ) +=b

a

c

b

c

adxxfdxxfdxxf

Cedxe xx +=

Ca

edxe

axax +=

( ) +=+ dvdudvdu

−= vduuvudv (Integration by Parts)

Ca

adxa

uu += ln

( ) ( )10ln

1 ≠>+= aaCa

ab

dxabx

bx

Caxaxx

dx += lnlnln

( ) ++=+

Cbxabbxa

dxln

1

( ) +±+=±

Caxxax

dx 22

22ln

+−=−

Cxaxa

xdx 22

22

+±=±

Caxax

xdx 22

22

Ca

x

aaxx

dx +=−

−

1

22sec

1

Cxxx

dx +=−

−1

2sec

1

( ) ( ) +

+=

+

Cmax

dxaxxm

mn

1ln

ln1

( )1≠m

( ) Caxae

dxxeax

ax +−= 12

+−= − Cdxexan

exa

dxex axnaxnaxn 11

( ) +−= Caxae

dxxeax

ax 12

−=ba

bxdxbx

axnaxn

ln

( ) ( )10ln

1 ≠>+ − bbCbxdxx

ban n

+=+

− Cxx

dx 12 tan

1

Cxa

axadx +=+

−122 tan

1

Cxaxa

axadx +

−+=

− ln21

22

( )022 >− xa

Caxax

aaxdx +

+−=

− ln21

22

( )022 >− ax

( ) Cxaaxa

xdx +±±=±

2222 ln

21

( ) Cx

axaaxx

dx ++−=+ ln

1

( ) ( )( ) ( )1

1

1

−≠++

+=++

nCnbbxa

dxbxan

n

( ) ( ) ++

−=+

Cbxabbxa

dx 12

( ) ( ) ++

==+

Cbxabbxa

dx23 2

1

=+ 222 xba

dx

Ca

xbabx

b+

++ 2

22

1ln1

=− dxxba 222

Cabx

ba

xbax +++ −1

2222 sin

22

+=−

− Cabx

bxba

dx 1

222sin

1

( ) Cbeaaca

xbea

dx cxcx

++−=+ ln

1

( ) ( )C

bxabbxa

dx ++

−=+ 34 3

1

( )[ ] ++−=+

Cbxaabxbbxa

xdxln

12

( )( ) +

+++

=+

Cbxabxa

abbxa

xdxln

122

( ) ( )( )( ) ++−

−=+ − C

bxabnbxa

dxnn 11

1 ,

( )1≠n

( ) ++=+ Cbxab

dxbxa 3

32

=− dxxa 22

Cax

axax +

+− −1222 sin21

( ) +−−=− Cxadxxax2/32222

31

=± dxax 22

( )[ ]Caxxaaxx +±+±± 22222 ln21

( ) Caxdxaxx +±=± 2/32222

31

=+=−

− Cax

xa

dx 1

22sin

( )0cos 221 >−+− − xaCax

=+=+

− C

xa

xa

dx 1

22sin

( ) Cxax +++ 22ln

=+=−

− Cax

ax

dx 1

22cosh

Caxx +−+ 22ln

−−=− − a

xdx

xxa 1

2

22

sin

Cx

xa +− 22

( )−++=+

222

22

ln xaxdxx

xa

Cx

xa ++ 22

Cxaax

axx

dx +−=− 2

22

222

( )Cx

xx

xd +=−

− −

1

2csc

1

−=

− abx

abxbadx 1

222 tan1 (x in rads)

( )0,0 >> ba

=+=−

− Cabx

abxbadx 1

222 tanh1

Cbxabxa

ab+

−+

ln2

1 (x in rads)

( ) ( ) ++−=

+ 223 22

bxab

bxa

bxa

xdx

( ) ( ) ++−=

+ 324 63

bxab

bxa

bxa

xdx

⋅=+ 3

2

21bbxa

dxx

( )( ) ( )[ ] Cbxaabxabxa ++++−+ ln23 2

( ) =+ 2

2

bxa

dxx

( ) ( ) Cbxaa

bxabxabx

b+

+−+

+ln2

213

( ) =+ 3

2

bxa

dxx

( )( )

( ) Cbxabxa

bxaab

+

++

++

ln2

43123

( ) ( ) ++++−=

+C

bxab

xbabxa

bxa

dxx33

222

4

2

333

++=+ 222222

2xba

xdxxba

Ca

xbabx

ba +

++ 2

222

1ln2

=+ bxax

dx2

Cbxax

dxa

bax

bxa ++

++− 2

( ) =+ 222 xa

dx

( ) Cax

axaax ++

+−1

3222 tan21

2

( ) =− 222 xa

dx

( ) Cxa

dxaxaa

x +−

+− 222222 2

12

( ) ++

=+

Caxa

x

ax

dx2222/322

( ) ++

−=+

Caxax

xdx222/322

1

( ) ±±=± 322222

4ax

xdxaxx

( ) Caxxa

axxa +±+−± 22

422

2

ln88

( ) +−−=− 322222

4xa

xdxxax

Cax

axaxa +

+− −1222

sin8

(a >0)

=+ dxbxax

( ) ( )

Cb

bxabxa+

+−−2

3

15322

=+ dxbxax 2

( ) ( )C

b

bxaxbabxa+

++−3

3222

105151282

( ) ++=+

Cabxbbxa

xdx 22 ln

21

?tan 1

2

2

+−=+

− Cabx

abb

abx

bxadxx

=++ 2cxbxa

dx

Cc

bcxcxbxa

c+

++++2

ln1 2

( )04

2sin

12

1 >+

−

−−−

= − cCacb

bcx

c

( ) Cbxaba

bx

bxaxdx ++−=+ ln

2

( ) ( )−+−=− 1

2222

naxx

dxaxn

( ) ( )11

222

2

−≠+−+

−nCdxax

nna n

( ) =−

nax

dx22

( )( ) ( ) +

−−−

− Cax

dxanaxx

n 222

2

22

2

( )2≠n

( ) ( )2

222

22 ++

−=− +

Cn

axdxaxx

nn

=− 222 xax

dx

Cx

xaaa

+++−22

ln1

=− 22 xax

xdx

Cxaxa

axa +−−

−− 21 2sin

( ) =+ dxbxax n

( )

Cn

an

bxabbxa n

+

+−

+++ +

122

1

,

( )2,1 −−≠n

( ) ( )C

nbxa

bdxbxa

nn

++

+=+ +

22

2

,

( )2−≠n

=+dx

xbxa

++

++ Cbxax

dxabxa2

tan2 1 +

−+

−=

+ − C

abxa

abxax

dx

ln1 +

++−+=

+ Cabxa

abxa

abxax

dx

=+dx

xbxa

2

Cbxax

dxbx

bxa ++

++− 2

+

−=− − a

axadxxaxx 1

32 sin

22

( )( )C

xaxaxax +−−+6

232 2

=− dx

xxax 22

Ca

axaxax +

−+− −12 sin2

=− dx

xxax

2

22

Ca

axx

xa +

−−−− −1sin2

2

( ) ++=+ 2222222 28

xaxax

dxxax

( ) Cxaxa +++− 22

4

ln8

=+dx

xxa 22

Cx

xaaaxa +++−+

2222 ln

=+ 22

2

xa

x

( ) Cxax

xaxa +++++

2ln

2

2222

2

++−=+

Cxa

xa

xax

dx2

22

222

==− −

axa

dxxax 14

222 sin8

( ) Cxaxax +−− 2222 281

=−dx

xxa 22

Cx

xaaxa +−+−−

2222 ln

+−−=−

Cxa

xadx

xa

x2

22

22

2

=− dxax 22

Caxxa

axx +−++− 22

222 ln

22

( ) += Cxdxx tansec2

( ) Cxdxx +−= cotcsc2

( ) +−= Cxdxx sinlncot 2

( ) Caaxx

dxax +−= 42sin

2sin 2

( ) ++= Caaxx

dxax42sin

2cos2

( ) ( ) Cxaxa

dxax +−= tan1

tan 2

( ) +−= Caxa

dxax cot1

csc2

( ) += Caxa

dxax tan1

sec2

=−dx

xax 22

Cxa

aax +−− −122 sec

=− dx

xax

2

22

Cx

axaxx +−−−+

2222ln

=−

dxax

x22

2

Caxx

axxa +−+−+ 2222

2

2ln

2

=− 22 axx

dx

Cxa

aC

ax

a+=+ −− 11 cos

1sec

1

Ca

ax

xax

dx +

−=−

−

1

2sin

2

=− 22 xax

dx

Ca

axaxax

ax +

−+−− −12

2 sin2

22

( ) ( )( ) +

−−=−1

22

22

nxaxax

dxxaxn

n

( ) +−+

+−

Cdxxaxnna n 2

22

21

( )( )( )

( ) +−

−−

−

−

2

22

2 22

2 anxaxax

xax

dxa

n

( )( ) ( ) +

−−−

− Cxax

dxan

nn 2

22

223

( ) = − dxax1cos

Cxaa

axx +−−− 221 11

cos

( ) =− dxax1tan

( ) Cxaaxx ++−− 221 1ln21

tan

( ) Ca

axxa

axdxaxx +−=

cossinsin 2

(x in rads)

( ) Ca

axxa

axdxaxx ++=

sincoscos 2

(x in rads)

Cx

xaaxaxx

dx +−−=−

21

2 2

( ) +−= Cxdux cossin

( ) += Cxdux sincos

( ) +−= Cxdxx coslntan

( ) ++−= Cxxdxx cotcsclncsc

( ) Cxxdxx ++= tanseclnsec

( ) += Cxdxx sinlncot

( ) +−= Caxa

dxax cos1

sin

( ) += Caax

dxaxsin

cos

( ) ( ) =+−= Caxa

dxax cosln1

tan

( ) Caxa

+secln1

( ) ( ) =+−= Caxaxa

dxax cotcscln1

csc

Cax

a+

2tan

1

( ) ( ) =++= Caxaxa

dxax tansecln1

sec

Cax

a+

+42

tan1 π

( ) ( ) Caxa

dxax += sinln1

cot

( ) Cxx

dxx +−= 42sin

2sin 2

( ) Cxx

dxx ++= 42sin

2cos2

( ) +−= Cxxdxx tantan 2

( ) ( )( ) −

++−=

baxba

dxbxax2

coscossin

( )( ) ( )22

2cos

baCba

xba ≠+−−

( ) ( )( ) −

−−−= ba

xbadxbxax

2sin

sinsin

( )( ) ( )22

2sin

baCba

xba ≠+++

( ) ( )( ) +

−−=

baxba

dxbxax2

sincoscos

( )( ) ( )22

2sin

baCba

xba ≠+++

( ) =+−−= Cxaxa

dxax cot1

cot 2

Caxa

+− sinln1

( ) ( ) +−=

−

naaxax

dxaxn

n cossinsin

1

( ) Cdxaxn

n n +−

−2sin1

( ) +=−

naaxax

dxaxn

n sincoscos

1

Caxn

n n +−

−2cos1

( ) =dxaxntan

( ) ( ) ( )1tan1

tan 21

≠+−−

−−

nCdxaxna

ax nn

( ) ( )( ) +

−=

−

1cotcsc

csc2

naaxax

dxaxn

n

( ) ( )1csc12 2 ≠+

−−

− nCdxaxnn n

( ) ( )( ) +

−=

−

1tansec

sec2

naaxax

dxaxn

n

( ) ( )1sec12 2 ≠+

−−

− nCdxaxnn n

( ) ( ) −−

−=−

1cot

cot1

naax

dxaxn

n

( ) ( )1cot 2 ≠+ − nCdxaxn

( ) = − dxax1sin

Cxaa

axx +−+− 221 11

sin

Caxaax

dx +−= cot1

sin 2

Caxaax

dx += tan1

cos2

+

+=−

Cax

aaxdx

24tan

1sin1

π

(x in rads)

Cax

aaxdx +

−−=+ 24

tan1

sin1π

(x in rads)

+−=−

Cax

aaxdx

2cot

1cos1

+=+

Cax

aaxdx

2tan

1cos1

( ) =dxaxx sin2

Caxaxa

axa

x ++− cos2

sin2

3

22

2

( ) = dxaxx cos2

Caxaxa

axa

x +−+ sin2

cos2

3

22

2

( ) =dxaxx n sin

( ) Cdxaxxan

axax n

n

++− − coscos 1

( ) = dxaxx n cos

( ) Cdxaxxan

axax n +−

− sinsin 12

( ) =− dxaxx n 1sin

Cxa

dxxn

aax

nx nn

+−+

−+

+−

+

22

11

1

11sin

1

( )1−≠n

( ) ++

= −+

− axnx

dxaxxn

n 11

1 cos1

cos

( )111 22

1

−≠+−+

+

nCxa

dxxn

a n

( ) −+

= −−

− axnx

dxaxxn

n 11

1 tan1

tan

( )111 22

1

−≠+−+

+

nCxa

dxxn

a n

( ) +−= Cxdxxx csccotcsc

( ) += Cxdxxx sectansec

( ) +−= Ca

axdxaxax

42cos

cossin

( ) += Caxa

dxax sinh1

cosh

( ) ( ) += Caxa

dxax coshln1

tanh

( ) += Cax

adxhax

2tanhln

1csc

( ) ( ) += − Caxa

dxhax tanhsin1

sec 1

( ) += Caxa

dxax sinhln1

coth

( ) +−= Cx

aax

dxax24

2sinhsinh 2

( ) ++= Cx

aax

dxax24

2sinhcosh 2

( ) +−= Caxa

xdxax tanh1

tanh 2

( ) ( ) ++

=+

Canax

dxaxaxn

n

1sin

cossin1

( )1−≠n

( ) ( ) ++

−=+

Canax

dxaxaxn

n

1cos

sincos1

( )1−≠n

( ) +−= Cna

axdxaxax

nn csc

cotcsc

( )0≠n

( ) Cna

axdxaxax

nn +=

sectansec

( )0≠n ( ) = dxaxax mn cossin

( )( ) +

+−

+−

nmaaxax mn 11 cossin

( ) ++− − Cdxaxax

nmn mn cossin

1 2

( )mn −≠ ( ) = dxaxax mn cossin

( )( ) +

+

−+

nmaaxax mn 11 cossin

( ) +++ − Cdxaxax

nmm mn 2cossin

1

( )nm −≠ ( ) =dxbxeax sin

( ) Cbxbbxaba

eax

+−+

cossin22

( ) =dxbxeax cos

( ) Cbxbbxaba

eax

+−+

sincos22

( ) =dxaxx sinh

Caxa

axxa +− sinh

1cosh 2

( ) =dxaxx cosh

Caxa

axxa +− cosh

1sinh 2

( ) =dxaxx n sinh

( ) Cdxaxxan

axax n

n

+− − coshcosh 1

( ) =dxaxx n cosh

( ) Cdxaxxan

axax n

n

+− − sinhsinh 1

=+ axcb

dxsin

−+−

−− −

24tantan

2 1

22

axcbcb

cba

π

( )22 cbC >+

=+ axcb

dxsin

axcbaxbcaxbc

bca sincossin

ln1 22

22 +−++

−−

( )22 bcC >+

=+ axcb

dxcos

Cax

cbcb

cba+

+−

−−

2tantan

2 1

22

( )22 cb >

=+ axcb

dxcos

Caxcb

axbcaxbc

bca+

+−++

− cossincos

ln1 22

22

( )22 cb < ( ) ( ) += Cxdxx coshsinh

( ) ( ) += Cxdxx sinhcosh

( ) ( ) += Cxdxx coshlntanh

( ) += Cx

dxhx2

tanhlncsc

( ) ( ) += − Cxdxhx tanhsinsec 1

( ) += Cxdxx sinhlncoth

( ) += Caxa

dxax cosh1

sinh

( ) +−= Caxa

dxaxh coth1

csc 2

( ) += Caxa

dxaxh tanh1

sec 2

( ) +−= Caxa

xdxax coth1

coth 2

( ) ( ) −=

−

naaxax

dxaxn

n cossinhsinh

1

( ) ( )0sinh1 2 ≠+−

− nCdxaxn

n n

( ) ( ) +=

−

naaxax

dxaxn

n sincoshcosh

1

( ) ( )0cosh1 2 ≠+−

− nCdxaxn

n n

( ) Cdxaxn += tanh

( ) ( ) Cdxaxanax n

n

++−

−−

21

tanh1

tanh

( )1≠n

( ) ( ) +−

−=−

anaxaxh

dxaxhn

n

1cothcsc

csc2

( ) ( )1csc12 2 ≠+

−−

− nCdxaxhnn n

( ) ( ) +−

=−

anaxaxh

dxaxhn

n

1tanhsec

sec2

( ) ( )1sec12 2 ≠+

−−

− nCdxaxhnn n

( ) ( ) +−

=−

anax

dxaxn

n

1coth

coth1

( ) ( )1coth 2 ≠+ − nCdxaxn

( ) Cna

axhdxaxaxh

nn +−=

sectanhsec

( )0≠n

( ) Cna

axhdxaxaxh

nn +−=

csccothcsc

( )0≠n ( ) =dxbxeax sinh

( )22

2baC

bae

baee bxbxax

≠+

−−

+

−

( ) =dxbxe ax cosh

( )22

2baC

bae

baee bxbxax

≠+

−+

+

−