# What is de Moivre's theorem?

de Moivre worked out a brilliant and beautiful way to solve complex equations.

If you for example have z3= 1 and you want to find all real and complex z that satisfy this equation. i is just a complex number written in rectangular form; z = 1 + i * 0 = cos(µ) + i sin(µ) . If you remember your specific angles for sine and cosine you want an angle that gets cosine(µ) = 0 and sine(µ)=1 so µ = 0+ 2nπ. Now, the + 2nπ is especially important for reasons you should see soon.

Rewrite i in euler form 1 = ei * (0 + 2nπ) = z3.  Now, if we take the cube root of both sides, that will be the same as taking it to the power of 1/3.

Remember your power rules: (ab)= ab * c. This will give you z = ei(0+n2π)/3. This is where the + 2nπ gets really important. If not, the answer would just be one. Put in the different values for n (n=0, n=1, n=2, etc) gives you your different angles. Plug them on your Argand diagram, and you get three different solutions:

z= 1 ; z= ei 2π/3 and z2 = ei 4π/3

Hope it helped. If not, well take it in the session

Answered by Frederik Dahl M. Maths tutor

172 Views

See similar Maths IB tutors
Maths tutor

172 Views

See similar Maths IB tutors Need help with Maths?

Have a Free Video Meeting with one of our friendly tutors. Need help with Maths?

Have a Free Video Meeting with one of our friendly tutors. Need help with Maths?

Have a Free Video Meeting with one of our friendly tutors.