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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