Using z=cos(θ)+isin(θ), find expressions for z^n-1/z^n and z^n+1/z^n

We make use of De Moivre's Theorem which states that (cos(θ)+isin(θ))^n=cos(nθ)+isin(nθ).z^n-1/z^n=cos(nθ)+ isin(nθ)-cos(-nθ)- isin(-nθ)=cos(nθ)+ isin(nθ)-cos(nθ)+ isin(nθ) (by trig relationships)=2isin(nθ)Similarly z^n+1/z^n=cos(nθ)+ isin(nθ)+cos(-nθ)+isin(-nθ)=cos(nθ)+ isin(nθ)+cos(nθ)- isin(nθ) (by trig relationships)=2cos(nθ)

BS
Answered by Bogosi S. Further Mathematics tutor

4954 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

Express the complex number (1+i)/(1-i) in the form x+iy


Prove that "6^n + 9" is divisible by 5 for all natural numbers.


Using a Taylor's series or otherwise; derive Euler's Formula


Solve this equation: x^2 + 2x + 2


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning