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Level 496

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Describe how to derive Polar Form or Modulus Argument Form

- Find r = |Z|, modulus, by √( (Im(Z))^2 + (Re(Z))^2 )

cos(x) + i sin (x)

What does cis stand for ?

z* = r cis (2 π - θ)

If z = r cis θ, what does z* equal

-z = r cis ( π + θ)

If z = r cis θ, what does -z equal

-z* = r cis ( π - θ )

If z = r cis θ, what does -z* equal

r₁r₂ cis (θ₁+θ₂)

In modulus argument form, z₁z₂ =

r₁/r₂ cis (θ₁ - θ₂)

In modulus argument form, z₁/z₂ =

r e ^(iθ)

Write z = r cis θ in Euler's form

Write z₁z₂ in Euler's form

r₁ r₂ e ^ ( i (θ₁ + θ₂) )

Write z₁ / z₂ in Euler's form

( r₁ / r₂ ) × e ^ ( i (θ₁ - θ₂) )

Write 1/z in Polar Form and Euler's Form

( 1 / r ) cis ^ ( - θ )

In Polar form, for Im(Z) = 0

θ = 0 or π or -π

In Polar form, for Re(Z) = 0

θ = π/2 or -3π/2 or 3π/2 or -π/2

θ = θ - 2π

If an angle is equal to θ in Argand diagram, the angle is also equal to ...

θ = π/2 - α

If tan θ = tan α, find θ.

If tan θ = - cot 2α, find θ.

θ = - ( π /2 - 2α )

If cis θ, find the for - cis

cis θ = - cis (θ + π)

If -cis θ, find cis

- cis θ = cis ( θ - π )

Find individual polar forms of nominator and denominator,

What to do when you cannot find the Polar form of a division of two complex numbers in Cartesian form

- Find z₁ × z₂ or z₁ / z₂, depending on question in cartesian and polar form

What to do for questions where the exact value of tan θ, sin θ, cos θ is required ?

What is a complex number?

An expression of the form

(a+bi)+(c+di)

Add the real parts and the imaginary parts

(a+bi)-(c+di)

Subtract the real parts and the imaginary parts

(a+bi)(c+di)=?

Multiply complex numbers like binomials, using i^2=-1

How do you simplify a complex number?

i, -1, - i, 1 divide the exponent by 4 and match the

What are the rules for the square roots of negative numbers?

Must have an "i" in the answer. ex. sqrt -36 = 6i