Formulas from Trigonometry

Formulas from Trigonometry: sin2 A + cos2 A = 1 cos(A ? B) = cos A cos B sin A sin B sin 2A = 2 sin A cos A

sin(A ? B) = sin A cos B ? cos A sin B

tan(A ? B)

=

tan A?tan B 1tan A tan B

cos 2A = cos2 A - sin2 A

tan 2A

=

2 tan A 1-tan2 A

cos

A 2

=

?

1+cos A 2

sin2 A

=

1 2

-

1 2

cos 2A

sin

A

+

sin

B

=

2

sin

1 2

(A

+

B)

cos

1 2

(A

-

B)

cos

A

+

cos

B

=

2

cos

1 2

(A

+

B)

cos

1 2

(A

-

B)

sin A sin B

=

1 2

{cos(A

-

B)

-

cos(A

+

B)}

sin A cos B

=

1 2

{sin(A

-

B)

+

sin(A

+

B)}

sin

A 2

=

?

1-cos A 2

tan

A 2

=

sin A 1+cos A

cos2

A

=

1 2

+

1 2

cos 2A

sin

A

-

sin

B

=

2

cos

1 2

(A

+

B)

sin

1 2

(A

-

B)

cos

A

-

cos

B

=

2

sin

1 2

(A

+

B)

sin

1 2

(B

-

A)

cos A cos B

=

1 2

{cos(A

-

B)

+

cos(A

+

B)}

cos() = sin( + /2)

Differentiation Formulas:

d dx

(uv)

=

u

dv dx

+

du dx

v

Chain rule:

dy dx

=

dy du du dx

d dx

cos

u

=

-

sin

u

du dx

d dx

sin-1

u

=

1 1-u2

du dx

,

-

2

<

sin-1 u

<

2

d dx

tan-1 u

=

1 1+u2

du dx

,

-

2

<

tan-1 u

<

2

d dx

ln u

=

1 du u dx

d dx

u v

=

v(du/dx)-u(dv/dx) v2

d dx

sin

u

=

cos

u

du dx

d dx

tan

u

=

sec2

u

du dx

d dx

cos-1 u

=

-1 1-u2

du dx

,

(0

<

cos-1 u

<

)

d dx

eu

=

eu

du dx

d dx

loga u

=

loga u

e

du dx

,

a

=

0, 1

Integration Formulas:

Integration by parts: u dv = uv - v du

du

= ln |u|

u

au du =

au ,

ln a

a > 0, a = 1

cos u du = sin u

sin2 u du

=

u 2

-

sin 2u 4

=

1 2

(u

-

sin

u

cos

u)

cos2 u du

=

u 2

+

sin 2u 4

=

1 2

(u

+

sin

u

cos

u)

du u2 - a2

=

1 2a

ln

u-a u+a

du

= ln(u + u2 + a2)

u2 + a2

eax sin bx dx

=

eax(a sin bx-b cos bx) a2+b2

x sin ax dx

=

sin ax a2

-

x cos ax a

sin2 ax dx

=

x 2

-

sin 2ax 4a

x2

cos ax dx

=

2x a2

cos ax

+

x2 a

-

2 a3

sin ax

tan2 ax dx

=

tan ax a

-x

ln x dx = x ln x - x

eu du = eu

sin u du = - cos u

tan u du = - ln cos u

tan2 u du = tan u - u

du u2 + a2

=

1 a

tan-1

u a

du

a2 - u2

= sin-1

u a

du

= ln(u + u2 - a2)

u2 - a2

eax cos bx dx

=

eax(a cos bx+b sin bx) a2+b2

x2

sin ax dx

=

2x a2

sin ax

+

2 a3

-

x2 a

cos ax

x cos ax dx

=

cos ax a2

+

x sin ax a

cos2 ax dx

=

x 2

+

sin 2ax 4a

xeax dx

=

eax a

x

-

1 a

x ln x dx

=

x2 2

ln

x

-

1 2

1

Rules for Exponents: ab+c = abac

(ab)c = abc

(ab)c = acbc

a-b =

1b 1 a = ab

Taylor Series: Euler's Formula:

ex = 1 + x + x2 + x3 + x4 + ? ? ? = xn

2! 3! 4!

n!

n=0

x2 x4 x6

(-1)nx2n

cos x = 1 - + - + ? ? ? =

2! 4! 6!

(2n)!

n=0

x3 x5 x7

(-1)nx2n+1

sin x = 1 - + - + ? ? ? =

3! 5! 7!

(2n + 1)!

n=0

ej = cos + j sin

ej + e-j cos =

2

ej - e-j sin =

2j

Rectangular and Polar Form of a Complex Number:

z = a + jb = rej

r = |z| = a2 + b2

a = Re{z} = r cos

r2 = zz b

= arctan a

z + z a = Re{z} =

2 b = Im{z} = r sin

z - z b = Im{z} =

2j

Phasors:

Complex Signal:

z(t) = Aej(0t+) = Aejej0t

Real Signal:

x(t) = Re{z(t)} = A cos(0t + )

Phasor Representation: X = Aej

Phasor Addition:

Let x1(t) = A1 cos(w0t + 1), x2(t) = A2 cos(w0t + 2), and x(t) = x1(t) + x2(t). Then x(t) = A cos(w0t + ) and:

the phasor representation for x(t) is X = Aej = A1ej1 + A2ej2.

Continuous-Time Unit Impulse and Unit Step:

(t) dt = 1

-

x(t)(t) dt = x(0)

- t

u(t) = (t) dt

-

x(t)(t - t0) dt = x(t0)

-

2

Discrete-Time Unit Impulse and Unit Step:

[n] =

1, n = 0, 0, otherwise.

u[n] =

1, n 0, 0, n < 0.

Complex Exponential Signals:

ej0t

ej0n

Distinct signals for distinct w0 Identical signals for values of w0 separated by multiples of 2

Periodic for any choice of w0

Periodic only if w0/(2) = m/N Q

Fundamental frequency w0

Fundamental frequency w0/m

Fundamental period: w0 = 0: undefined

w0 = 0: 2/w0

Fundamental period: w0 = 0: one

w0 = 0: N = 2m/w0

Periodicity of Discrete-Time Sinusoids:

cos(0n), sin(0n), and If periodic, then write

ej0n are periodic in reduced form:

if and only if w0 is a ratio of

w0

=

m

(no

2 common

factors

two integers. between m and

N)

2 N

N : Fundamental Period

m: In each period of the discrete-time signal, the graph "goes around" m times.

Summation Formulas:

N2

k

=

N1

-

N2+1 ,

1-

k=N1

=1

ak =

1 ,

1-a

k=0

|a| < 1

kak =

a ,

(1 - a)2

k=0

|a| < 1

n

ak

=

1

-

an+1 ,

1-a

k=0

a=1

n

kak

=

a{1

-

(n + 1)an + (1 - a)2

nan+1}

k=0

Time Domain Representation of Discrete-Time Signals:

x[n] = ? ? ? + x[-2][n + 2] + x[-1][n + 1] + x[0][n] + x[1][n - 1] + x[2][n - 2] + ? ? ?

=

x[k][n - k].

k=

Systems:

System H is linear if H{ax1[n] + bx2[n]} = aH{x1[n]} + bH{x2[n]}.

System H is time invariant if H{x[n - n0]} = y[n - n0].

Impulse response: for LTI system H, h[n] = H{[n]}.

3

Convolution:

y[n] = x[n] h[n] =

x[k]h[n - k] =

x[n - k]h[k]

k=-

k=-

Convolution with [n]:

x[n] [n] = x[n]

x[n] [n - n0] = x[n - n0].

4

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