Formula Sheet

Pythagorean Identity

sin2 θ + cos2 θ = 1

Sum and Difference Formulas

sin(α+ β) = sinα cosβ + cosα sinβ

sin(α− β) = sinα cosβ − cosα sinβ

cos(α+ β) = cosα cosβ − sinα sinβ

cos(α− β) = cosα cosβ + sinα sinβ

Double-Angle Formulas

sin(2α) = 2 sinα cosα

cos(2α) =

cos2(α)− sin2(α)2 cos2(α)− 11− 2 sin2(α)

Euler’s Formula

ejθ = cos θ + j sin θ

cos θ =1

2(ejθ + e−jθ)

sin θ =1

2j(ejθ − e−jθ)

a+ jb = R 6 θ

R =√a2 + b2

θ = tan−1(b

a

)a = R cos θ

b = R sin θ

Multiplying and Dividing Phasors

You can just multiply by complex conjugate or do this

R1 6 θ1 ·R2 6 θ2 = (R1 ·R2)6 θ1 + θ2

R1 6 θ1R2 6 θ2

=

(R1

R2

)6 θ1 − θ2

Useful Stuff

dn

dsn

(1

s+ a

)=

n!

(s+ a)n+1

(s2 + 1) = (s+ j)(s− j)

Impulse ResponseJust find inverse Laplace of the transfer functionUnit Step ResponseJust integrate the inverse Laplace of the Transfer Function

Unit Ramp ResponseYou can either integrate the step response or L −1

{H(s)s2

}Polynomial Long Divison

1

x2 − x− 2)

x2

− x2 + x+ 2

x+ 2

Complex Root Expansion

1. Factor the polynomial into complex parts

2. Multiply by one of the polynomials and solve for thevalue you chose

3. The other value will just be the conjugate of what youfound

4. Inverse Laplace given by

2|k1|eαt cos(βt+ 6 k1)

where α = Re{numerator} and β = Im{numerator}

Multiple Root Expansion

1. Multiply by the multiple root.

2. Plug in the value of s to make the double root = 0.

3. Solve for kn = 10! (H(s) ·multiple root)|s

4. kn−1 = 1(1)!

d1

ds1 (H(s) ·multiple root)|s

5. kn−2 = 1(2)!

d2

ds2 (H(s) ·multiple root)|s

6. kn−3 = 1(3)!

d3

ds3 (H(s) ·multiple root)|s7. . . .

8. k1 = 1(n−1)!

d(n−1)

ds(n−1) (H(s) ·multiple root)|s

Complex Exponential Partial Fraction Expansion

1. Split up the transfer function into

normal

denominator+

exponential

denominator

2. The exponential part is just a time domain shift of

1

denominator

3. Solve both parts individually and add to get the finalanswer

1