Studied by 37 peoplemodulus-argument form
z = r(cosθ + isinθ) where |z| = r arg z = θ
exponential form
z=re^iθ
rules for multiplying/dividing complex numbers in mod-arg/exponential form
multiply moduli, add arguments (or the reverse)
what does the locus |z-(x+iy)|=n represent
a circle of radius n, with centre (x,y)
what does the locus |z-(x+iy)|=|z-(a+ib)| represent
the perpendicular bisector of the line between (x,y) and (a,b)
what does the locus arg(z-(x+iy))=θ represent
a "half line" starting from (x,y) and going at the angle θ from the positive real axis
first step in finding the min/max modulus of circle loci
find the line that goes through the centre and the origin
first step in finding min/max argument of circle loci
draw a tangent to the circle that passes through the origin (considering negative arguments)
how to find min modulus of the perpendicular bisector loci
find the modulus of the intersection point between the locus line and the line perpendicular to that which passes through the origin
how to eliminate -isinθ
change both θs into (-θ)
exponential form
re^iθ
θ in radians
argument of a real number
-π, 0, π
argument of imaginary number
π/2, -π/2
expression for cosθ
expression for sinθ
derive the expression for cosθ from Euler’s relation
derive the expression for sinθ from Euler’s relation
how to derive a formula for sin(nθ) and cos(nθ)
cos(nθ)+isin(nθ) = (cosθ+isinθ)ⁿ (de Moivre’s theorem)
expand this bracket
the real parts are cos(nθ) and the imaginary parts are sin(nθ)
how to change cosⁿx/sinⁿx into sum of multiple angles
express trig function in exponential form
let z=e^iθ such that cosⁿ(x) = ½[z+(1/z)] and sin(x)=(1/2i)[z-(1/z)]
expand this using binomial expansion
gather inverse terms together
zⁿ±(1/zⁿ)= 2cos(nθ)/2sin(nθ) - de Moivre
done!
integral of cosnθ
(1/n)[sin(nθ)]
integral of sin(nθ)
(-1/n)cosnθ
Note
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