How do I express y=acosx+bsinx in the form y=Rcos(x-c)?

From the addition formula, we know that:

Rcos(x-c) = Rcos(x)sin(c)+Rsin(x)cos(c)


acos(x)+bsin(x) = Rcos(x)cos(c)+Rsin(x)sin(c)

If we equate the coefficients of cos(x) and sin(x) we see that:

acos(x) = Rcos(x)sin(c);   therefore a = Rcos(c)

And that:

bsin(x) = Rsin(x)cos(c);    therefore b = Rsin(c)

To find c:

If we divide one of the above results by the other:

Rsin(c)/Rcos(c) = b/a

Rsin(c)/Rcos(c) = b/a

tan(c) = b/a

Therefore, c = arctan(b/a)

To find R:

a2+b2 = R2cos2(c)+R2sin2(c)

a2+b2 = R2(cos2(c)+sin2(c))

As cos2(c)+sin2(c) = 1,

a2+b2 = R2


So, overall:

acos(x)+bsin(x) = (a2+b2)1/2cos(x-arctan(b/a))

Daniel R. A Level Maths tutor, A Level Further Mathematics  tutor, GC...

1 year ago

Answered by Daniel, an A Level Maths tutor with MyTutor

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


£20 /hr

Matthew M.

Degree: Engineering Mathematics (Bachelors) - Bristol University

Subjects offered: Maths, Law+ 1 more


“I am an Engineering Math student at the University of Bristol. I have a range of interests, which cover maths, history and law, and would love to help others in these areas as well as encourage the same enthusiasm I have in others! I a...”

MyTutor guarantee

£20 /hr

Seb G.

Degree: Mathematics (Masters) - Bath University

Subjects offered: Maths


“Me: I'm a second year maths student at the University of Bath. I fell in love with maths after a rocky start with the subject, so believe that I can help people regardless of whether they like maths or not.  I've been tutoring maths l...”

MyTutor guarantee

£20 /hr

Jed M.

Degree: Natural Sciences (Bachelors) - Exeter University

Subjects offered: Maths, Chemistry+ 1 more


“Hi! I'm a second year Natural Science student at the University of Exeter. I love all things science, and enjoy sharing what I know about it with others. For my GCSEs and A Levels, I found worked examples and practice questions are ke...”

About the author

£24 /hr

Daniel R.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Physics+ 1 more

Further Mathematics

“About me I’m a first year student studying Maths with European Studies at Durham. I have recently taken my A levels, achieving A* in Maths and Further Maths, so I am familiar with the course content and what the examiners are looking ...”

You may also like...

Other A Level Maths questions

Integrating cos^2(x)+5sin^2(x)

A curve has equation y = 6ln(x) + x^2 -8x + 3. Find the exact values of the stationary points.

A rollercoaster stops at a point with GPE of 10kJ and then travels down a frictionless slope reaching a speed of 10 m/s at ground level. After this, what length of horizontal track (friction coefficient = 0.5) is needed to bring the rollercoaster to rest?

Find the solution(s) of 3(x^2)-6x+2=0

View A Level 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