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...

10 months ago

Answered by Ellie, an IB Maths tutor with MyTutor

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


£36 /hr

James G.

Degree: Mathematical Physics (Doctorate) - Nottingham University

Subjects offered:Maths, Physics+ 2 more

Further Mathematics

“Currently a 3rd year PhD student in Mathematical Physics. I'm very passionate about teaching as well as my subject area. Look forward to hearing from you.”

MyTutor guarantee

£30 /hr

Michelangelo M.

Degree: MMath Mathematics with Placement (Masters) - Bath University

Subjects offered:Maths, Italian+ 4 more

Further Mathematics

“Mathematics student at the University of Bath, willing to help you love this subject and improve your grades.”

£26 /hr

Tadas T.

Degree: MMathPhil Mathematics and Philosophy (Bachelors) - Oxford, St Anne's College University

Subjects offered:Maths, Further Mathematics + 3 more

Further Mathematics
-Personal Statements-
-Oxbridge Preparation-

“University of Oxford Maths and Philosophy student happy to help students learn and stay motivated!”

About the author

Ellie S.

Currently unavailable: for new students

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

Find the area under the curve of f(x)=4x/(x^2+1) form x = 0 to x = 2.

An arithmetic sequence has third term 56 and 15th term 92. What is the common difference and the first term?

In a lottery, 6 numbered balls are drawn from a pool of 59. Calculate the probability of scoring a jackpot. There used to be 49 balls in the pool. Calculate by how much the addition of 10 balls has decreased the probability of scoring a jackpot

Take the square root of 2i

View IB Maths tutors

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