write the sum cos(x)+cos(2x)+...+cos(nx) as a quotient only involving sine and cosine functions

We can write this sum S as Re(e^ix+e^2ix+...+e^nix), we now have a finite geometric series, which we know the formula for.Have, S = Re( e^ix(1-e^inx)/(1-e^ix)) - Now factoring numerator and denominator to look like complex formula for sine function we get,S = Re( e^ixe^inx/2(e^-inx/2-e^inx/2)/(e^ix/2(e^-ix/2-e^ix/2))) = Re(e^i(n/2+1/2)xsin(nx/2)/sin(x/2))Now since n is an integer and x is an element of the reals taking the real part gives,S = sin(nx/2)cos(((n+1)/2)x)/sin(x/2)

Answered by Further Mathematics tutor

4344 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

Given that f(x)=2sinhx+3coshx, solve the equation f(x)=5 giving your answers exactly.


A complex number z has argument θ and modulus 1. Show that (z^n)-(z^-n)=2iSin(nθ).


Solve the equation 2(Sinhx)^2 -5Coshx=5, giving your answer in terms of natural logarithm in simplest form


Find the eigenvalues and eigenvectors of the matrix M , where M{2,2} = (1/2 2/3 ; 1/2 1/3) Hence express M in the form PDP^-1 where D is a diagonal matrix.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences