exponential form
(r)e^iĪø
argument of a real number
-Ļ, 0, Ļ
argument of imaginary number
Ļ/2, -Ļ/2
expression for cosĪø
expression for sinĪø
derive the expression for cosĪø from Eulerās relation
derive the expression for sinĪø from Eulerās relation
how to derive a formula for sin(nĪø) and cos(nĪø)
cos(nĪø)+isin(nĪø) = (cosĪø+isinĪø)āæ (de Moivreās theorem)
expand this bracket (let cosĪø=c and sinĪø=s for ease of layout)
the real parts are cos(nĪø) and the imaginary parts are sin(nĪø)
how to change cosāæx/sināæx into an integrable form and what is this form
form is sum of multiple angles
express trig function in exponential form
let e^iĪø = z so that
cosāæ(Īø) = (Ā½)āæ[z+(1/z)]āæ
sināæ(Īø)=(1/2i)āæ[z-(1/z)]āæ
expand this using binomial expansion
gather inverse terms together
zāæĀ±(1/zāæ)= 2cos(nĪø)/2isin(nĪø) - de Moivre
done!
how to turn a combination of sin and cos into an integrable form
same procedure as above, but include difference of two squares to make calculations easier
integral of cos(nĪø)
(1/n)sin(nĪø)
integral of sin(nĪø)
(-1/n)cosnĪø
when to employ āthe trickā
when there is a 1Ā±e^iĻ
what is āthe trickā
for 1Ā±e^iĻ, multiply the whole expression by e^(-iĻ/2) over itself
what to do when something isnāt ātrickableā
for nĀ±k(e^iĻ), multiply by nĀ±k(e^-iĻ) [ie the complex conjugate] over itself
how to find sums of trig series for cos and sin, given one of the series
find the corresponding cos/sin series
find cos+i sin
convert into exponential form to form geometric series
apply formula then manipulate it (using trick or non-trick)
sum of cos series is real part/ sum of sin series is imaginary part
what to do for trig series with binomial coefficients
C + iS
convert to exponential form (still with binomial coefficients)
figure out what bracket is being expanded
apply āthe trickā to the bracket and manipulate
apply the power to each term in the bracket
cos series is real part/ sin series is imaginary part
the nth roots of unity are
1, e^i(2Ļ/n), e^i(4Ļ/n),ā¦ā¦, e^i(2(n-1)Ļ/n)
how to find the nth roots of a number (two different ending methods) ie how to solve zāæ=w
express equation as rāæe^inĪø=Re^iĻ
solve rāæ=R (r must be positive as r=|z|)
solve nĪø=Ļ
either solve up to nĪø=Ļ+2(n-1)Ļ
or let Ļ=e^i2Ļ/n and x=āāR e^iĻ/n, then the roots are x, xĻ, xĻĀ²ā¦.xĻāæā»Ā¹