ECE 3364 Circuits II Power Formulas (front) Instantaneous power )(*)()( titvtp = In AC-circuits

)cos()( 0 vm tvtv += and )cos()( 0 im titi += Inductors: 2/= iv . Capacitors: 2/= iv [ ] avtivtivtp ivmmivmm += )2sin()sin(2)2cos(1)cos(2)( 00 where )(ti is the reference0 [ ] avtivtp ivivmm +++= )2cos(1)cos(2)( 0 where iv & can be referenced with respect to any reference0

Average power wattsiv

P ivmm )cos(

2 = Reactive power sivQ ivmm var)sin(2 =

Power factor angle ivpfa = Power factor )cos( ivpf = Reactive factor )sin( ivrf = Lagging (current-phase lags voltage-phase) iv > leading (current-phase leads voltage-phase) iv < RMS values

2m

rmsv

v = and 2m

rmsi

i = so that wattsivP ivrmsrms )cos( = Note: "rms" and "effective" are synonymous, so you often see effv instead of rmsv , etc.

Complex power avjQPS += Apparent power avQPS += 22 ZIZVIVS rmsrmsrmsrms2*2* ~/~~~ ===

aviv

eiv

S ivmmmm iv == )(

22)( wattsSP iv )cos( = sSQ iv var)sin( =

Given S , a lagging pf , wattspfSP ||= and spfSQ var))(cossin( 1+= Given S , a leading pf , wattspfSP ||= and spfSQ var))(cossin( 1= In phasor notation (AC-circuits)

vmvV =~ and imiI =~ or in RMS units vrmsvmrms vvV ==2

~ and irmsi

mrms ii

I ==2

~

( ) LoadrmsLoad

rmsrmsrms

ivmm ZI

ZVavIVIV

ivS 2

*

2**|~|

)(|~|~~

2

~~)(

2=====

Balanced 3-phase circuits (AC) Pos phase seq: ( ) ANBN VV ~1200.1~ = ; ( ) ANCN VV ~1200.1~ += and [ ] ABBC VV ~1200.1~ = ; [ ] ABCA VV ~1200.1~ += Neg phase seq: ( ) ANBN VV ~1200.1~ += ; ( ) ANCN VV ~1200.1~ = and [ ] ABBC VV ~1200.1~ += ; [ ] ABCA VV ~1200.1~ = Pos phase seq: ( ) ANAB VV ~303~ += aAAB II ~30

31~

+= note: aAAN II ~~ =

Neg phase seq: ( ) ANAB VV ~303~ = aAAB II ~303

1~

=

wattsPPIVP loadYCloadY

BIVrmsaA

rmsAN

loadYA aAAN

=== )cos(~~ ATTL PP 3= sQQIVQ loadYC

loadYBIV

rmsaA

rmsAN

loadYA aAAN var)sin(

~~ === ATTL QQ 3= ( ) CBIVrmsaArmsANrmsaArmsANA SSIVIVS aAAN ==== )(~~~~ *

Voltage source transformation: Y and Y Y : ( ) naab vv '' 303 o+= YZZ 3= Y : '' 30

31

abo

na vv

= = ZZY 3

1

For Y load transformation, just let voltages =0 For Y load transformation, just let voltages=0 +-

+-

+ -

Z a

a

bc

vca

vbc

vab

b

c +- +-

+-+-

+ -+ -

Z Z a

a

bc

vcavca

vbcvbc

vabvab

b

c

+-

+- +-

Z Ya

a

b

c

vann

vbnvcn

+-+-

+-+- +- +-

Z YZ Ya

a

b

c

vanvann

vbnvbnvcnvcn

ECE 3364 Circuits II Power Formulas (back) Currents/Voltages at the load Design of a load impedance LLL XRZ j+= for maximum power transfer to the load. Case 1: Both LR and LX can be chosen without any constraints. Choose *)( thL ZZ = . Case 2: Constrained. bLRa and dLXc . Step 1: Choose LX as near to

*)( thL XX = as dLXc permits. Step 2: choose LR as near to ( ) ( )22 LththL XXRR ++= as bLRa permits.

Case 3: LZLL ZZ = where LZ is specified and cannot be changed, and LZ can be chosen by the circuit designer. Choose thL ZZ = Design of a series-impedance ddd jXRZ += for maximum power transfer to the load. Case 1: Both dR and dX can be chosen without any constraints. Choose 0=dR . Choose dX so that ( ) 0=++ LXdXthX Case 2: Constrained. bRa d and dXc d . Step 1: choose dR as near to 0 as bdRa permits. Step 2: Choose dX so that ( )LXdXthX ++ is as near to 0 as dXc d permits. Case 3: dZdd ZZ = where dZ is specified and cannot be changed, and dZ can be chosen by the circuit designer. Choose [ ]dd ZLthZLthd XXRRZ sin)(cos)( +++= if it yields a value 0>dZ , else choose

0== dd XR Wattmeters

= ivrmsrms IVreadingwattmeter cos~~ 21 WWPTTL += 13 WQTTL =

+-V th Z =R +jXL L L

thZ =R +jXth th Z =R +jXd d d

+-V thV th Z =R +jXL L LZ =R +jXL L L

thZ =R +jXth ththZ =R +jXth th Z =R +jXd d dZ =R +jXd d d

+-V th Z =R +jXL L L

thZ =R +jXth th

+-V thV th Z =R +jXL L LZ =R +jXL L L

thZ =R +jXth ththZ =R +jXth th

Positive phase sequence

30

30

30

VCA VAB

VBC

VCN

VBN

VAN30

3030

I CA

I AB

I BC

I cC

I bB I aA

Positive phase sequence

30

30

30

VCA VAB

VBC

VCN

VBN

VAN

30

30

30

VCAVCA VABVAB

VBCVBC

VCNVCN

VBNVBN

VANVAN30

3030

I CA

I AB

I BC

I cC

I bB I aA

30

3030

I CA

I AB

I BC

I cC

I bB I aA

Negative phase sequence

30

30

30

I CA

I AB

I BC

I cC

I bB I aA

VCA VAB

VBC

VCN

VBN

VAN

30

3030

Negative phase sequence

30

30

30

I CA

I AB

I BC

I cC

I bB I aA30

30

30

I CA

I AB

I BC

I cC

I bB I aA

VCA VAB

VBC

VCN

VBN

VAN

30

3030

VCAVCA VABVAB

VBCVBC

VCNVCN

VBNVBN

VANVAN

30

3030

A

B

C

3phaseload

cc2

cc1pc1

pc2

+

+

+

+

A

B

C

3phaseload

cc2

cc1pc1

pc2

++

++

++

++

A

B

C

3phaseload

cc1

pc1

+

+

A

B

C

3phaseload

cc1

pc1

++

++

ECE 3364 Circuits II Formula Sheet: Inductance and Transformers

Self-inductance: dttdtvL)()( = where )()()( tiPNtNt LL == td

tidLtdtidPN

dttdtv LLL

)()()()( 2 === :)(t flux-linkage; )(tL : magnetic flux; P : permeance of space occupied by the flux; N : number of wire turns

Mutual inductance dot convention for windings: Current into dots causes flux that adds. Use right-hand-rule for flux-direction. dot convention for transformer circuit: current into the dot of a transformer coil causes a voltage drop across the other coils with positive polarity at the dot in the other coils.

21LLkM = coefficient of coupling 0.10 k ; 2

1

2

1LL

NN =

coils not magnetically linked -> k=0; all flux from each coil links other coil -> k=1.0

)()( 1,21,12

11 PPNL += )()( 2,12,2222 PPNL += For non-magnetic core: 1,22,1 PP = )()()( 111,21,11 tiNPPt += )()()( 222,12,22 tiNPPt +=

dttidPPN

dttdNtv )()()()()( 11,21,1

21

111 +== ; dt

tidPPNdt

tdNtv )()()()()( 22,12,22

22

22 +== Phasor domain: Linear transformer model

2

2line22

2line2

*2line22

22

reflected2ndary

)()(

)(

LL XXLRRR

ZLjRMZ

+++++++=

22line2

22line2

22

reflectedL )()()(

LL

LL

XXLRRRjXRMZ +++++

=

111line

reflected LjRZZVMj

Vs

s+++

= ;

2

11line2

11line

11line11line22

)()(

)]()[(

LXXRRR

LXXjRRRMZ

ss

ssrefl

+++++

++++=

Phasor domain: Ideal transformers ; Reflect to primary side: Reflect to secondary side

i (t)1 i (t)2

L1 L 2 v (t)2 v (t)1

+

-

+

-

N1 N 2i (t)1 i (t)1 i (t)2 i (t)2

L1 L1 L 2 L 2 v (t)2 v (t)2 v (t)1

+

-v (t)1 v (t)1

+

-

+

-

N1N1 N 2N 2

v (t) 1 v (t) 2

i (t) 1 i (t) 2 ++

- -

M

v (t) 1 v (t) 2

i (t) 1 i (t) 2 ++

- -v (t) 1v (t) 1 v (t) 2

i (t) 1 i (t) 2 ++

- -v (t) 2v (t) 2

i (t) 1i (t) 1 i (t) 2i (t) 2 ++

- -

M

Zs

+- V s ZL

N1N 2

2ZsZs

+- V s +- V s V s ZLZL

N1N1N 2N 2

2+-V s ZL

1 ZsN2 N2

N2N1

+-V s V s ZLZL1 ZsN2 N

21 ZsN2 N

2ZsZsN2N2 N

2N2N1N2N2N1

Zs

+- V s ZL

N1 N 2ZsZs

+- V s +- V s V s ZLZL

N1N1 N 2N 2

)()( 11

22 tvN

Ntv =)()( 12

12 tiN

Nti =

jwM

jwL1

R1 R2

I 1 I 2 ZL

Zline 1

jwL2

Zline 2

+-

Zs

V s

jwM

jwL1 jwL1

R1 R1 R2 R2

I 1 I 1 I 2 I 2 ZLZLZL

Zline 1Zline 1

jwL2 jwL2

Zline 2Zline 2Zline 2

+-

Zs

V s +-+-

ZsZs

V s V s

jwL1 R1

I s

Zline 1

ZL reflectedZ2ndary reflected

+-

Zs

V s

jwL1 jwL1 R1 R1

I s I s

Zline 1Zline 1

ZL reflectedZL reflectedZ2ndary reflectedZ2ndary reflected

+-+-

ZsZs

V s V s

R2

I 2 ZL

jwL2 Zline 2

+-

Zreflected

V reflected

R2 R2

I 2 I 2 ZLZL

jwL2 jwL2 Zline 2Zline 2

+-+-

ZreflectedZreflected

V reflectedV reflected

Reflect to primary side

Reflect to secondary side

jwM

jwL1

R1 R2

I 1 I 2 jwL2

V 1 V 2 +

-

+

-

jwM

jwL1 jwL1

R1 R1 R2 R2

I 1 I 1 I 2 I 2 jwL2 jwL2

V 1 V 1 V 2 V 2 +

-

+

-( )( ) ( )( )( )( ) ( )( )22212

21111ILjRIMjVIMjILjRV

++=++=

Phasor domain: 3-phase ideal transformers

Ideal three-phasetransformer

A

b Z line

c Z line

a

Z line+-

a

n b

c

+

-

+-

Z S-YZ S-Y

Z S-YvS-Y~

vcn~ vbn~

B

CZ L-Y

N

Z L-Y

Z L-Y

-YY-

Y-Y

N1 N 2

Ideal three-phasetransformer

A

b Z lineZ line

c Z lineZ line

a

Z lineZ line+-+-

a

n b

c

+

-

+

-

+- +-

Z S-YZ S-YZ S-YZ S-Y

Z S-YZ S-YvS-Y~vS-YvS-Y~

vcn~vcnvcn~ vbn~vbnvbn~

B

CZ L-YZ L-Y

N

Z L-YZ L-Y

Z L-YZ L-Y

-YY-

Y-Y

-YY-

Y-Y Y-Y

N1 N 2N1N1 N 2N 2

-Y

Y-

Y-Y

and

Reflected to primary Reflected to secondary

+-Z S-Y

vS-Y~Z L-Y

N1N 2

2Z line

Z L-Y+-

N1

N 22Z S-Y

N1

N 2 vS-Y~

N1

N 22Z line

+-Z S-Y

vS-Y~3 Z L-Y

N1N 2

2Z line

+-Z S-Y

vS-Y~Z line

3 Z L-YN1N 2

21

Z L-Y+-

N1

N 22Z S-Y N1

N 22Z line3

131

N1

N 2 vS-Y~31 -30

3

Z L-Y+-

N1

N 22Z S-Y N1

N 22Z line3

vS-Y~N1N 2+303

-Y

Y-

Y-Y

and

Y-Y

and

Reflected to primary Reflected to secondary

+-Z S-Y

vS-Y~Z L-Y

N1N 2

2Z line+-+-

Z S-YZ S-Y

vS-Y~vS-YvS-Y~Z L-YZ L-Y

N1N 2

2N1N1N 2N 2

2Z lineZ line

Z L-Y+-

N1

N 22Z S-Y

N1

N 2 vS-Y~

N1

N 22Z line

Z L-YZ L-Y+-+-

N1

N 22Z S-YN1

N 22

N1N1

N 2N 22Z S-YZ S-Y

N1

N 2 vS-Y~N1N 2N1N1

N 2N 2 vS-Y~vS-YvS-Y~

N1

N 22Z lineN1

N 22

N1N1

N 2N 22Z lineZ line

+-Z S-Y

vS-Y~3 Z L-Y

N1N 2

2Z line+-+-

Z S-YZ S-Y

vS-Y~vS-YvS-Y~3 Z L-Y

N1N 2

23 Z L-YZ L-Y

N1N 2

2N1N1N 2N 2

2Z lineZ line

+-Z S-Y

vS-Y~Z line

3 Z L-YN1N 2

21+-+-

Z S-YZ S-Y

vS-Y~vS-YvS-Y~Z lineZ line

3 Z L-YN1N 2

213 Z L-YZ L-Y

N1N 2

2N1N1N 2N 2

21

Z L-Y+-

N1

N 22Z S-Y N1

N 22Z line3

131

N1

N 2 vS-Y~31 -30

Z L-YZ L-Y+-+-

N1

N 22Z S-YN1

N 22

N1N1

N 2N 22Z S-YZ S-Y N1

N 22Z lineN1

N 22

N1N1

N 2N 22Z lineZ line3

131

3131

N1

N 2N1N1

N 2N 2 vS-Y~vS-YvS-Y~3131 -30

3

Z L-Y+-

N1

N 22Z S-Y N1

N 22Z line3

vS-Y~N1N 2+303

3

Z L-YZ L-Y+-+-

N1

N 22Z S-YN1

N 22

N1N1

N 2N 22Z S-YZ S-Y N1

N 22Z lineN1

N 22

N1N1

N 2N 22Z lineZ line3

vS-Y~vS-YvS-Y~N1N 2+303 N1N 2N1N1

N 2N 2+303

Phasor domain: 3-phase linear transformers Reflect to primary side: Reflect to secondary side: Y-Y transformer Y-Y transformer Y- transformer Y- transformer -Y transformer -Y transformer

N

R1 Zline 1

Zrefl 1+-

Zs

jwL1

n

Aa

van~IaA~

N

R1 R1 Zline 1Zline 1

Zrefl 1+-+-

ZsZs

jwL1 jwL1

n

Aa

van~vanvan~IaA~IaA~

Linearthree-phasetransformer

A

b Z line 1

c Z line 1

a

Z line 1+-

a

n b

c

+

-

+-

Z S-YZ S-Y

Z S-YvS-Y~

vcn~ vbn~

B

CZ L-Y

N

Z L-Y

Z L-Y

-YY-

Y-Y

jwM

Z line 2

Z line 2

Z line 2

Linearthree-phasetransformer

A

b Z line 1Z line 1

c Z line 1Z line 1

a

Z line 1Z line 1+-+-

a

n b

c

+

-

+

-

+- +-

Z S-YZ S-YZ S-YZ S-Y

Z S-YZ S-YvS-Y~vS-YvS-Y~

vcn~vcnvcn~ vbn~vbnvbn~

B

CZ L-YZ L-Y

N

Z L-YZ L-Y

Z L-YZ L-Y

-YY-

Y-Y

-YY-

Y-Y Y-Y

jwMjwM

Z line 2Z line 2

Z line 2Z line 2

Z line 2Z line 2

211line

211line

11line11line22

)()(

)]()[(

LXXRRR

LXXjRRRMZ

ss

ssrefl

+++++

++++=

111linereflected

~

LjRZZVMjV

YSYS

+++=

Z line2Z L-Y

a A

+-1

3-30 o Vrefl

~

Z refl3

jwL2 3

R2 3

Z line2Z line2Z L-YZ L-Y

aa AA

+-+-1

3-30 o Vrefl

~13

-30 o13

1

33-30 o-30 o Vrefl

~Vrefl~

Z refl3

Z reflZ refl3

jwL2 3

jwL2 jwL2 33

R2 3

R2 R2 33

( )222

21 33 LjRZZ

MZYLline

refl

+++=

jwL1 R1

I s

Zline 1

ZL reflectedZ2ndary reflected

+-

Zs

V s

jwL1 jwL1 R1 R1

I s I s

Zline 1Zline 1

ZL reflectedZL reflectedZ2ndary reflectedZ2ndary reflected

+-+-

ZsZs

V s V s

22line2

22line2

22

reflectedL )()()(

LL

LL

XXLRRRjXRMZ +++++

=

R2

I 2 ZL

jwL2 Zline 2

+-

Zreflected

V reflected

R2 R2

I 2 I 2 ZLZL

jwL2 jwL2 Zline 2Zline 2

+-+-

ZreflectedZreflected

V reflectedV reflected

111linereflected LjRZZ

VMjV

s

s+++

=

211line

211line

11line11line22

)()(

)]()[(

LXXRRR

LXXjRRRMZ

ss

ssrefl

+++++

++++=

22line2

22line2

*2line22

22

reflected2ndary

)()(

)(

LL XXLRRR

ZLjRMZ

+++++++=

N

1 3 R1 Zline 1

Zrefl 1+-

Zs

n

Aa

van~IaA~

jwL1 1 3

N

1 3 R1 1 3 1 3 R1 R1 Zline 1Zline 1

Zrefl 1+-+-

ZsZs

n

Aa

van~vanvan~IaA~IaA~

jwL1 1 3 jwL1 jwL1 1 3 1 3

( )111line

'refl 33

303~LjRZZ

VMjV

sna

+++

=o

( ))(3 222

21 LjRZZ

MZYLline

refl

+++= ( )

)33 111

22 LjRZZ

MZlines

refl

+++=

Z line2Z L-Y

a A

+- Vrefl~

Z refl 2 jwL2 R2 Z line2Z line2Z L-YZ L-Y

aa AA

+-+- Vrefl~Vrefl~

Z refl 2Z refl 2 jwL2 jwL2 R2 R2

Table of 1-sided Laplace Transforms { } == + 0

)()()( dtetxtxLsX st .

)(tx )(sX

[ ] nn dttd /)( ns )(tu 1s

!)(

ntuet tn ( ) 1

1++ ns

)()cos( 0 tut 20

2 +ss

)()sin( 0 tut 20

20

+

s

( ) )()cos(exp 0 tutt ( ) 202 +++

ss

( ) )()sin(exp 0 tutt ( ) 2020

++

s

[ ] )()cos()sin( 000 tuttt ( )2202302

+

s

)()sin( 00 tutt ( )2202202

+

s

s

( ) )()sin(exp 00 tuttt ( )( )[ ]2202202

+++

s

s

( )[ ] )()cos()sin(exp 000 tutttt ( )[ ]2202302

++

s

( ) )()sin()cos(exp tutbbaABtbAat

22)( bas

BAs++

+

( ) )()cos(exp)!1(

2 1tutbat

rtK

K

r+

Krr KK

b

jbasK

jbasK =

++++

0

)()(

*

Properties of 1-sided Laplace Transforms 1. Linearity )()()]()([ sbYsaXtbytaxL +=+ 2. Laplace transform of the derivative

)0()0(...)0()0()0()()()()1(

)1(

)2(

)2(

2

2321

=

n

n

n

nnnnn

n

n

dtxd

dtxds

dtxds

dtdxsxssXstu

dttxdL

3. Integration =+=

0|

)()()(t

t

dttdx

ssXdxL

4. s-shift )()]([ asXtxeL at += and )]([)]([ 11 sXLeasXL at =+ 5. Delay 0)()]()([ >= asXeatuatxL as 6. Convolution theorem: { } )()()()()()()(*)(

00sYsXduutyuxLduuyutxLtytxL

tt =

=

=

7. Product { } )(*)()()( sYsXtytxL = . 8. Time scaling { } 01)( >

= aforasX

aatxL

9. Initial value theorem (provided the limit exists) 10. Final value theorem (provided the limit exists) )]([lim)(lim

0ssXtx

st +=+ )]([lim)(lim 0 ssXtx st + = if and only if all poles

of )(ssX lie in the left half closed s-plane.

Circuit Response Impulse response: circuit output when all initial conditions equal to zero and input equals a Dirac delta function. Step response: circuit output when all initial conditions equal to zero and input equals a step function. Natural response: due only to initial conditions, voltage/current sources are cancelled. Transient response: portion of the output that goes to zero as time tends to infinity. Steady-state response: portion of the output that does not go to zero as time tends to infinity. Complete response: circuit output due to input and any non-zero initial conditions. LaPlace-domain Circuit Analysis Damping of 2nd-order Circuits

2 poles: 0012 =++ asas yields poles 0212/12,1 4)(5.0 aaas = )

2nd-order RLC circuit: the parameters 01, aa are functions of RLC, can be designed to have either of three dampings. Two real and equal poles 2,1s : critically damped Two real and distinct poles 2,1s : over-damped Two complex conjugate poles 2,1s : under-damped

s-domain inductor and capacitor Transfer Function s-domain Change the circuit to the s-domain. RR , sLL , , )/(1 sCC set all initial conditions set to 0. Set the s-domain source to value 1.0, label the s-domain output as H(s) and use KVL or KCL to compute H(s). Impulse Response The impulse response )(th of the circuit is { })()( sHILTth = . Frequency Response The frequency response )(H of the circuit is jssHH == |)()( .

LL

+

-

L i(0 ) +-C

v (0 )Cs

CLL

L

+

-

+

-

+

-

L i(0 ) +-+-C

v (0 )Cs

C

LaPlace domain solutions: Linear Transformers with non-zero initial conditions time-domain Linear transformer model Note: possible non-zero initial conditions

S-domain Linear transformer model th

thth sCsLRsZ 1)(1 ++=

L

LLL sCsLRsZ 1)( ++=

s

viLiMiLsIC thcth

)0()0()0()0()( 12111

++=

s

viLiMiLsIC LcL

)0()0()0()0()( 21222

+++=

Reflect to primary side: )(

)(22

22

sZsLRMssZ

Lrefl ++

=

)(

)()(22

2refl sZsLR

sICMssVL++

=

Reflect to secondary side: ; )(

)(111

22

sZsLRMssZrefl ++

=

[ ]

)()()(

)(1111

refl sZsLRsICsVMs

sV s +++=

sL1 R1

I (s) 1 Z (s)1

+- V (s) s

+

- IC (s) 1

Z (s)refl

+- V (s) refl

sL1 sL1 R1 R1

I (s) 1 I (s) 1 Z (s)1Z (s)1

+-+- V (s) s V (s) s

+

-

+

- IC (s) 1 IC (s) 1

Z (s)reflZ (s)refl

+-+- V (s) reflV (s) refl

+-

+

M

L1 L 2

R1 Rth L th Cth R2 RL L L

CL

-

v (t) th

v (t) C thi (t) 1 +

-Lv (t) C

i (t) 2 +-+-

+

M

L1 L1 L 2 L 2

R1 R1 RthRth L thL th CthCth R2 R2 RL RL L L L L

CL CL

-

v (t) thv (t) th

v (t) C thv (t) C thi (t) 1 i (t) 1 +

-Lv (t) C Lv (t) C v (t) C

i (t) 2 i (t) 2

sM

sL1

R1 R2

I (s) 1 I (s) 2 Z (s)LZ (s)1

+- V (s) s sL2

+-IC (s) 1 IC (s) 2 +-

sM

sL1 sL1

R1 R1 R2 R2

I (s) 1 I (s) 1 I (s) 2 I (s) 2 Z (s)LZ (s)LZ (s)1Z (s)1

+-+- V (s) s V (s) s sL2 sL2

+-+-IC (s) 1 IC (s) 1 IC (s) 2 IC (s) 2 +-+-

R2

Z (s)LZ (s)refl

+-V (s) refl

sL2 IC (s) 2 + -

R2 R2

Z (s)LZ (s)LZ (s)reflZ (s)refl

+-+-V (s) reflV (s) refl

sL2 sL2 IC (s) 2 IC (s) 2 + -+ -

A Brief Table of Integrals

= )()()()()()( xduxvxvxuxdvxu = aaxxdxax

4)2sin(

2)(sin 2

+=+1

1

nxdxxn

n += )sin()cos()cos( xxxdxxx

11

)1(>

= nfor

nxdxx

nn = )cos()sin()sin( xxxdxxx

( ) +=+ baxadxbxa ln11 [ ] )cos(2)sin(2)sin( 22 xxxxdxxx =

( ) >+=+

1)1(

)( )1( nfornbbxadxbxa

nn [ ] )sin(2)cos(2)cos( 22 xxxxdxxx +=

= aadxax

xln

== xxdxx seclncosln.tan

= axax eadxe )1( )tan()sec(ln1)sec( axaxadxax +=

= axax ea

axdxxe 21 == )sin(ln)csc(ln)cot( xxdxx

complexorrealaeaa

xaxdxex axax

+= 322

22 = )tan(1)(sec2 axadxax

[ ] ++= 22)sin()cos()cos( baeaxbbxadxebxax

ax

= )sec()tan()sec( xdxxx

= )csc()cot()csc( xdxxx

Some Trigonometric Identities: Reciprocal

csc1sin = sec

1cos = cot1tan =

sin1csc = cos

1sec = tan1cot =

Quotient

=

cossintan

=sincoscot

Pythagorean sin2a + cos2a = 1 tan2a + 1 = sec2a 1 + cot2a = csc2a Even-Odd Identities

)sin()sin( = )cos()cos( = )tan()tan( =

)csc()csc( = )sec()sec( = )cot()cot( = Sum & Difference

cossincossin)sin( = sinsincoscos)cos( m=

tantan1tantan)tan( m

= Double-Angle

cossin22sin = =

tan1tan2)2tan(

2 )(sin211)(cos2)(sin)(cos)2cos( 2222 ===

Half-Angle

)sin(5.0)2/cos()2/sin( =

2cos1

2sin =

2cos1

2cos +=

+

=

cos1cos1

2tan

Function-Product

)cos(5.0)cos(5.0sinsin += )cos(5.0)cos(5.0coscos ++=

)sin(5.0)sin(5.0cossin ++= )cos(5.0)sin(5.0sincos +=