Use De Moivre's Theorem to show that if z = cos(q)+isin(q), then (z^n)+(z^-n) = 2cos(nq) and (z^n)-(z^-n)=2isin(nq).

De Moivre's Theorem states that if z = cos(q)+isin(q), then

zn = (cos(q)+isin(q))n = cos(nq)+isin(nq)

But then 

z-n = cos(-nq)+isin(-nq).

Now, cos(-p)=cos(p), as cosine is a symmetric (even) function, and sin(-p)=-sin(p), as sine is an anti-symmetric (odd) fuction. Thus,

z-n  = cos(nq)-isin(nq).

The rest is just algebra:

zn+z-n = [cos(nq)+isin(nq)]+[cos(nq)-isin(nq)] = 2cos(nq).

zn-z-n = [cos(nq)+isin(nq)]-[cos(nq)-isin(nq)] = 2isin(nq).

Dorian A. A Level Physics tutor, A Level Maths tutor, A Level Further...

1 year ago

Answered by Dorian, an A Level Further Mathematics tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£26 /hr

Vandan P.

Degree: Physical Natural Sciences (Bachelors) - Cambridge University

Subjects offered: Further Mathematics , Science+ 6 more

Further Mathematics
-Oxbridge Preparation-

“I am a second year undergraduate at Corpus Christi College Cambridge. I am studying Natural Sciences. In my first year I did Physics, Chemistry, Computer Science and Maths and passed with first class honours....”

£26 /hr

George B.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Further Mathematics , Maths+ 1 more

Further Mathematics

“Third year undergraduate at one of the top universities for Maths. Eager to tutor and help improve your grades.”

£20 /hr

Rhoda A.

Degree: Electrical and Electronic Engineering (Masters) - Sheffield University

Subjects offered: Further Mathematics , Maths

Further Mathematics

“A Bit About MeI have just finished my second year at The University of Sheffield, studying Electrical and Electronic Engineering.I have always wanted to teach maths, this was because I had a great teacher (my dad). I believe that e...”

About the author

£24 /hr

Dorian A.

Degree: Theoretical Physics (Masters) - Durham University

Subjects offered: Further Mathematics , Physics+ 1 more

Further Mathematics

“About Me As a Theoretical Physics student at Durham University, I am more than aware of all of the confusing turns that science can take. I have areal passion for my subject, and hope to show my students howbeautiful science can be.  ...”

You may also like...

Posts by Dorian

A satellite is in a stationary orbit above a planet of mass 8.9 x 10^25 kg and period of rotation 1.2 x 10^5 s. Calculate the radius of the satellite's orbit from the centre of the planet.

Two lines have equations r = (1,4,1)+s(-1,2,2) and r = (2,8,2)+t(1,3,5). Show that these lines are skew.

Use De Moivre's Theorem to show that if z = cos(q)+isin(q), then (z^n)+(z^-n) = 2cos(nq) and (z^n)-(z^-n)=2isin(nq).

Other A Level Further Mathematics questions

How to approximate the Binomial distribution to the Normal Distribution

How do you find the cube root of z = 1 + i?

How do you differentiate arctan(x)?

What modules have you done before?

View A Level Further Mathematics tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss