Y1, C2 - Argand Diagrams Flashcards
What goes on the axis of an argand diagram
y = imaginary (Im)
x = real (Re)
What is the modulus of a complex number
The distance from the origin on an argand diagrams
What is the argument of a complex number
The angle the modulus makes with the positive real axis
The anti-clockwise rotation in radians
What is the range for the principal argument
-pi < x < pi
If you have an imaginary number in the negative y quadrants, how do you calculate the argument
pi - arctan(y / x)
What is the modulus argument form of an imaginary number
z = r(cosθ + isinθ)
Where r = modulus
θ = argument
When multiplying complex numbers, what happens to the modulus and argument of the result
The moduli multiply
The arguments add together
When dividing complex numbers, what happens to the modulus and argument of the result
The moduli are divided
The arguments are subtracted
What do you do if a complex number is in the form r(cosθ - isinθ)
Take the negative of the argument and turn the sin function to positive +
( rcos(θ) + isin(-θ) )
What is cos-θ equal to
cosθ
What is -sin-θ equal to
sinθ
What does it mean if a function is even
f(x) = f(-x)
Symmetrical in y axis
What does it mean if a function is odd
f(x) = -f(-x)
Rotational symmetry of order 2
What should you do if your value for θ is outside of your principal value
Subtract or add 2pi to it until it falls within the range
With loci, what does z denote
A general complex number
What does lz - z1l = r mean?
Circle, radius r
Centre z1
What does lz - z1l mean in relation to vectors
lz - z1l is the distance between z and z1
What is the cartesian equation of the locus lz - 5 - 3il = 3
z = x + iy
l (x-5) + (y-3)i l = 3
Find modulus:
root( (x-5)^2 + (y-3)^2 ) = 3
Square both sides:
(x-5)^2 + (y-3)^2 = 9
What is the centre and radius of a locus: l 4i + 2 - z l = 4
= l - (z - 2 - 4i) l = 4
= l z - (2 + 4i) l = 4
Centre = 2,4
Radius = 4
How do you find the minimum and maximum values of arg z
Find the tangent
Create a triangle with the radius
Use trigonometry
How do you find the minimum and maximum values of lzl
Work out the distance from 0 to centre
z max = distance + radius
z min = distance - radius
What does lz - z1l = lz - z2l mean?
The complex number z must be equal distance from both z1 and z2
How would you write y = 3 in complex number form
z = x + 3i ???
After finding the perpendicular bisector between two points (loci), how do you find the least possible value of lzl
Find the point perpendicular from the origin on the bisector and then find the intersection of the perpendicular bisector and the perpendicular line from the origin
How would you represent arg(z) = pi/6 on a graph
HALF-LINE from the origin angled at pi/6 anticlockwise direction
How do you draw a half line
Open circle at the end which is not included
Find the cartesian equation of arg(z + 3 + 2i) = 3pi / 4
tan(3pi / 4) = (y + 2) / (x + 3)
-1 = (y + 2) / (x + 3)
-x - 3 = y + 2
y = -x - 5
For x < -3
How would you find the complex number z that satisfies both lz + 3 + 2il = 10 and arg(z + 3 + 2i) = 3pi / 4
Cartesian equation:
(x+3)^2 + (y+2)^2 = 10^2
y = -x - 5
z = -3-5root(2) + (-2 + 5root(2))i
OR
Use coordinate geometry:
Draw out diagram and use triangles
How do you find the range from pi to -pi for values of theta where an argument and a circle have no common solutions
Create a kite with the distance between the centre of the circle and the half line point as well as the radius of the circle and then find theta
When should you use dotted and full lines with inequalities
Dotted if < or >
Full line if <= or >=
a) Find the cartesian equation of the locus of z if l z - 3 l = l z + i l. b) Hence find the least possible value of lzl
a) Using the definition of the modulus:
l x + iy - 3 l = l x + iy + i l
root((x-3)^2 + y^2) = root(x^2 + (y+1)^2)
-6x + 9 = 2y + 1
y = -3x + 4
b) (perp line) 1/3 * x = -3x + 4
x = 6/5 therefore y = 2/3
z = 6/5 + i2/5
l z lmin = 2root(2/5)
