exponential form
(r)e^iθ
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 (let cosθ=c and sinθ=s for ease of layout)
the real parts are cos(nθ) and the imaginary parts are sin(nθ)
how to change cosⁿx/sinⁿx into an integrable form and what is this form
form is sum of multiple angles
express trig function in exponential form
let e^iθ = z so that
cosⁿ(θ) = (½)ⁿ[z+(1/z)]ⁿ
sinⁿ(θ)=(1/2i)ⁿ[z-(1/z)]ⁿ
expand this using binomial expansion
gather inverse terms together
zⁿ±(1/zⁿ)= 2cos(nθ)/2isin(nθ) - 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(nθ)
(1/n)sin(nθ)
integral of sin(nθ)
(-1/n)cosnθ
when to employ “the trick”
when there is a 1±e^iϕ
what is “the trick”
for 1±e^iϕ, multiply the whole expression by e^(-iϕ/2) over itself
what to do when something isn’t “trickable”
for n±k(e^iϕ), multiply by n±k(e^-iϕ) [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 “the trick” 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(2π/n), e^i(4π/n),……, e^i(2(n-1)π/n)
how to find the nth roots of a number (two different ending methods) ie how to solve zⁿ=w
express equation as rⁿe^inθ=Re^iϕ
solve rⁿ=R (r must be positive as r=|z|)
solve nθ=ϕ
either solve up to nθ=ϕ+2(n-1)π
or let ω=e^i2π/n and x=ₙ√R e^iϕ/n, then the roots are x, xω, xω²….xωⁿ⁻¹