Find cos4x in terms of cosx.

As with any question involving sines and cosines, consider complex numbers as a likely way to find the answer. For this particular kind of question, where sin/cos of a multiple of x is needed in terms of sinx/cosx, or vice versa, the complex number representation z = cosx + isinx must be expanded in two ways. 

Firstly, expand using de Moivre's Theorem:

z = cosx + isinx

z4 = (cosx + isinx)4 =  cos4x + sin4x

Then, expand using the binomial expansion formula to get powers of cosx:

z4 = (cosx + isinx)4 = cos4x + 4icos3xsinx + 6i2cos2xsin2x + 4i3cosxsin3x + i4sin4x

                               = cos4x + 4icos3xsinx - 6cos2xsin2x - 4icosxsin3x + sin4x

Equate the real parts of both expansions to get cos equivalence:

cos4x = cos4x - 6cos2xsin2x + sin4x

Use cos2x + sin2x = 1 as a substitution:

cos4x = cos4x - 6cos2x(1-cos2x) + (1-cos2x)2

          = 8cos4x - 8cos2x + 1

Ellie S. IB Maths tutor, GCSE Maths tutor, IB Physics tutor, GCSE Phy...

4 months ago

Answered by Ellie, an IB Maths tutor with MyTutor

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


PremiumChristopher S. A Level Maths tutor, GCSE Maths tutor, A Level Spanish...
View profile
£26 /hr

Christopher S.

Degree: Mathematics (Masters) - Bristol University

Subjects offered: Maths, Spanish+ 1 more

Further Mathematics

“ I am extremely passionate about mathematics and I love the Spanish language! ”

Kyle R. GCSE Maths tutor, IB Maths tutor, A Level Maths tutor, Uni Ad...
View profile
£20 /hr

Kyle R.

Degree: Mathematics (Masters) - Oxford, Corpus Christi College University

Subjects offered: Maths, Further Mathematics + 2 more

Further Mathematics

“Hi! I'm Kyle, a 2nd year Oxford undergraduate studying Maths at Corpus Christi College. I aim to be an understanding and friendly face, whilst being thorough with the material I cover. I've acheived A* grades in Maths, Further Maths a...”

MyTutor guarantee

Jonas P. IB Physics tutor, A Level Physics tutor, GCSE Physics tutor,...
View profile
£22 /hr

Jonas P.

Degree: MPhys Theoretical Physics (Bachelors) - Edinburgh University

Subjects offered: Maths, Physics


“I am currently a student at the University of Edinburgh, doing my 4th year of the MPhys Theoretical Physics programme. I've managed to score an average grade of 81% in year 3, and I tend to consistently do well in my exams, which I be...”

About the author

Ellie S. IB Maths tutor, GCSE Maths tutor, IB Physics tutor, GCSE Phy...
View profile

Ellie S.

Currently unavailable: until 05/12/2016

Degree: Physics (Bachelors) - Oxford, Lady Margaret Hall University

Subjects offered: Maths, Physics+ 3 more

Further Mathematics
-Personal Statements-

“My Background: Hi! I'm Ellie, I'm 19, and I'm a Physics student at Oxford University. My strengths aremaths and physics, although I did a number of different subjects in sixth form, so I am also very familiar with Latin, English, and ...”

MyTutor guarantee

You may also like...

Posts by Ellie

A cricketer throws a ball vertically upwards so that the ball leaves his hands at a speed of 25 m/s. Calculate the maximum height reached by the ball, the time taken to reach max. height, and the speed of the ball when it is at 50% max. height.

Differentiate y = e^(x^2 - 3x).

Find cos4x in terms of cosx.

The tip of each prong of a tuning fork emitting a note of 320Hz vibrates in SHM with an amplitude of 0.50mm. What is the speed of each tip when its displacement is zero?

Other IB Maths questions

How do you calculate the probability P(X < x) for a normally distributed random variable X?

What does differentiation actually mean?

When do you use 'n choose k' and where does the formula come from?

Differentiation from first principles

View IB Maths tutors


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