Complex Numbers

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1/2i (e^inθ - e^-inθ)
Sin nθ
1 of 11
1/2 (e^inθ + e^-inθ)
Cos nθ
2 of 11
(r(Cos θ +iSin θ))^n
r^n(Cos nθ + iSin nθ)
3 of 11
z + 1/z =
z + z^-1 =
2cos θ
4 of 11
z - 1/z =
z - z^-1 =
2iSin θ
5 of 11
z^n + 1/z^n =
z^n + z^-n =
2cos nθ
6 of 11
z^n - 1/z^n =
z^n - z^-n =
2iSin nθ
7 of 11
For questions which ask for cos nθ = ... or sin nθ = ...
Use de Moivre's Theorem + binomial expansion
Compare real and imaginary parts
8 of 11
For questions which ask for cos^n θ = ... or sin^n θ = ...
let z = Cos θ +iSin θ
cos^n θ = (z + z^-1)^n
sin^n θ = (z - z^-1)^n
Binomially expand bracket
Group like terms
Replace like terms with 2Cosn θ OR 2iSin nθ
9 of 11
Sin nθ =
1/2i (e^inθ - e^-inθ)
10 of 11
Cos nθ =
1/2 (e^inθ + e^-inθ)
11 of 11

Other cards in this set

Card 2

Front

1/2 (e^inθ + e^-inθ)

Back

Cos nθ

Card 3

Front

(r(Cos θ +iSin θ))^n

Back

Preview of the front of card 3

Card 4

Front

z + 1/z =
z + z^-1 =

Back

Preview of the front of card 4

Card 5

Front

z - 1/z =
z - z^-1 =

Back

Preview of the front of card 5
View more cards

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