Complex Numbers
- How do I find a power of any complex number, like (1 + i√3)⁶, using De Moivre's theorem? In one line Convert the number to polar form r(cosθ+isinθ) ; then its n th power is simply rⁿ(cos nθ+isin nθ) . Explained simply Multiplying... 1
- How do I find all the nth roots of a complex number and why are there exactly n of them? In one line Write the number as r[cos(θ+2kπ)+isin(θ+2kπ)] and apply De Moivre's theorem with power 1/n ; taking k=0,1,dots,n-1 gives the n d... 1
- How do I solve equations like x⁴ + 1 = 0 using roots of unity? In one line Rewrite the equation as xⁿ= a complex number, express that number in polar form with 2kπ added, and take the n th roots. Explain... 1
- What are the Cartesian, polar and exponential forms of a complex number and how do I convert between them? In one line A complex number can be written as x+iy (Cartesian), r(cosθ+isinθ) (polar) or re^iθ (exponential); all three describe the same p... 1
- How do I add, multiply and divide complex numbers and use modulus, argument and conjugate? In one line Complex numbers follow ordinary algebra with the single extra rule i²=-1 ; the conjugate bar z=x-iy is the tool that removes i f... 1
- What does De Moivre's theorem state and how does it simplify powers of cos θ + i sin θ? In one line De Moivre's theorem says (cosθ+isinθ)ⁿ=cos nθ+isin nθ : raising this number to the power n simply multiplies the angle by n . Ex... 1
- How do I expand cos nθ and sin nθ in powers of cos θ and sin θ? In one line Write cos nθ+isin nθ=(cosθ+isinθ)ⁿ , expand the right side by the binomial theorem, and compare real and imaginary parts. Explai... 1
- How do I express powers like cosⁿθ and sinⁿθ in terms of cosines and sines of multiples of θ? In one line Put x=cosθ+isinθ , so 2cosθ=x+1/x and 2isinθ=x-1/x ; raise to the power, expand, and pair terms into x^k±frac1x^k . Explained si... 1
- Expand \(\sin^5\theta\cos^3\theta\) in a series of sines of multiples of \(\theta\). Let x=cosθ+isinθ : x-1/x=2isinθ,\ x+1/x=2cosθ,\ xⁿ-frac1xⁿ=2isin nθ . (2isinθ)⁵(2cosθ)³=(x-1/x)⁵(x+1/x)³=(x-1/x)²(x²-frac1x²)³ =(x²-2+frac1x... 1
- If \(\cos\alpha+\cos\beta+\cos\gamma=0,\ \sin\alpha+\sin\beta+\sin\gamma=0\), prove that (i) \(\cos3\alpha+\cos3\beta+\cos3\gamma=3\cos(\alpha+\beta+\gamma)\) (ii) \(\sin3\alpha+\sin3\beta+\sin3\gamma=3\sin(\alpha+\beta+\gamma)\) Let a=cosα+isinα,\ b=cosβ+isinβ,\ c=cosγ+isinγ . Then a+b+c=(cosα+cosβ+cosγ)+i(sinα+sinβ+sinγ)=0 . Identity: if a+b+c=0 then a³+b³+c³=3abc (... 1
- Find all the roots of \(\left(\frac12+i\frac{\sqrt3}{2}\right)^{3/4}\), hence find the continued product of the roots. Step 1 (polar form): 1/2+isqrt3/2=cosπ/3+isinπ/3 . Including the general angle, 1/2+isqrt3/2=cos(2kπ+π/3)+isin(2kπ+π/3) . Step 2 (De Moivre)... 1
- Solve \(x^6+1=0\). x⁶=-1=cosπ+isinπ=cos(2k+1)π+isin(2k+1)π . By De Moivre: x=cos(2k+1)π/6+isin(2k+1)π/6,\ k=0,1,dots,5 . k=0:\ fracsqrt32+i/2 ; k=1:\ i ; k=2:\... 1
- Prove that \(\sqrt{1+\mathrm{cosec}(\theta/2)}=\left(1-e^{i\theta}\right)^{-1/2}+\left(1-e^{-i\theta}\right)^{-1/2}\). Step 1: 1-e^iθ=e^(iθ/2)(e^(-iθ/2)-e^(iθ/2))=e^(iθ/2)(-2isinfracθ2)=2sinfracθ2 e^(i(fracθ2-fracπ2)) (since -i=e^(-iπ/2) ). therefore(1-e^iθ)^... 1
- If \(z=\frac1{\sqrt2}+\frac i{\sqrt2}\) and \(\bar z\) is the conjugate of z, then prove that \((z)^{10}+(\bar z)^{10}=0\). z=cosfracπ4+isinfracπ4, bar z=cosfracπ4-isinfracπ4 . By De Moivre: z¹⁰=cosfrac10π4+isinfrac10π4 , bar z¹⁰=cosfrac10π4-isinfrac10π4 . z¹⁰+bar... 1
- If \(\sin^4\theta\cos^3\theta=a_1\cos\theta+a_3\cos3\theta+a_5\cos5\theta+a_7\cos7\theta\), then prove that \(a_1+9a_3+25a_5+49a_7=0\). Let x=cosθ+isinθ : x-1/x=2isinθ,\ x+1/x=2cosθ,\ xⁿ+frac1xⁿ=2cos nθ . (2isinθ)⁴(2cosθ)³=(x-1/x)⁴(x+1/x)³=(x-1/x)(x²-frac1x²)³ =(x-1/x)(x⁶-3x²... 1
- Find all the values of \(\left(\frac12+i\frac{\sqrt3}2\right)^{3/4}\) and show that their continued product is 1. Step 1 (polar form): 1/2+isqrt3/2=cosπ/3+isinπ/3 . Including the general angle, 1/2+isqrt3/2=cos(2kπ+π/3)+isin(2kπ+π/3) . Step 2 (De Moivre)... 1
- Solve the equation \(x^4+x^3+x^2+x+1=0\). Multiply by (x-1) : (x-1)(x⁴+x³+x²+x+1)=x⁵-1=0 , with x≠1 . So the required roots are the fifth roots of unity other than 1: x⁵=1=cos2kπ+isi... 1
- Prove that \(\cos^6\theta=\frac{1}{32}(\cos6\theta+6\cos4\theta+15\cos2\theta+10)\). Let x=cosθ+isinθ . Then x+1/x=2cosθ and xⁿ+frac1xⁿ=2cos nθ . (2cosθ)⁶=(x+1/x)⁶=x⁶+6x⁴+15x²+20+15/x²+frac6x⁴+frac1x⁶ (binomial coefficients 1... 1
- Find all values of \((1+i)^{1/3}\). Polar form: 1+i=sqrt2(cosπ/4+isinπ/4)=sqrt2[cos(2kπ+π/4)+isin(2kπ+π/4)] . De Moivre: (1+i)^(1/3)=2^(1/6)[cos(8k+1)π/12+isin(8k+1)π/12],\ k=0... 1