Pure 2: Argand Diagrams Flashcards
What is one radian?
The measurement of an angle subtended by an arc of length r
How many radians in 360°?
2π
How many radians in 180°?
π
How many radians in 90°?
π/2
How many radians in 45°?
π/4
How many radians in 60°?
π/3
How many radians in 30°?
π/6
How many radians in 270°?
3π/2
How are complex numbers represented on Argand diagrams?
z=x+iy = (x,y)
What do complex conjugate pairs look like on Argand diagrams?
Reflections of each other in the x-axis
What happens when two points are added together on an Argand diagram?
The resulting complex number forms a parallelogram with the two added complex numbers and (0,0)
What is the modulus of a complex number?
The magnitude of its corresponding vector
How is the modulus calculated?
|z| = ✔️(x^2 + y^2)
What is the argument of a complex number? (2)
The angle (θ) that the vector makes with the positive real axis Usually given in the range -π < θ ≤ π (principal argument)
What is the modulus-argument form of a complex number?
z = r(cos(θ) +i sin(θ)) where r is the modulus and theta is the argument
How do you rearrange into the modulus-argument form?
For any θ:
sin(-θ) = -sin(θ)
cos(θ) = cos(-θ)
How are two modulus-argument forms multiplied? (2)
Multiply the modulus
Add the arguments
How are two modulus-argument forms divided? (2)
Divide the modulus
Subtract the arguments
What does |z - z1| represent on an Argand diagram?
A circle centre (x1, y1) with radius r
How is the modulus converted into a Cartesian equation?
|x+iy| = ✔️(x^2 + y^2) so |z|^2 = x^2 + y^2
How is the maximum and minimum modulus found? (4)
|z| means distance from (0,0)
Find distance from (0,0) to centre (using Pythagoras)
Maximum = distance to centre + radius
Minimum = distance to centre - radius
How is the maximum and minimum argument of r=|z - z1| found? (3)
Draw the circle with the locus of points
Draw a tangent from (0,0) to make the biggest and smallest angle
Use trigonometry to calculate the angle
What does |z - z1| = |z-z2| represent on an Argand diagram?
The perpendicular bisected of the line segment joining z1 to z2
How is the least possible value of |z| found given that |z - z1| = |z-z2|? (2)
Look for the perpendicular line that passes through (0,0)
Solve the simultaneous equations to find the complex number that is on both lines and calculate its modulus
What does ((z - z1) = θ) represent on an Argand diagram?
A half line from, but not including, the fixed point z1 that makes an angle θ with a line parallel to the real axis
How is the Cartesian equation of ((z - z1) = θ) found?
y - y1 = tan(θ)(x-x1)
