MYTUTOR SUBJECT ANSWERS

1348 views

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)

Therefore:

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

(a2+b2)1/2=R

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...

2 years ago

Answered by Daniel, an A Level Maths tutor with MyTutor


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

253 SUBJECT SPECIALISTS

£20 /hr

Iebad A.

Degree: Aerospace Engineering (Masters) - Sheffield University

Subjects offered:Maths, Italian

Maths
Italian

“About me: Hello, my name is Iebad and I am an aerospace engineering student at The University of Sheffield. As an engineer, maths is the key to unlock all the solutions to our problems in this field. Since the early years I had a broad...”

£22 /hr

Sam F.

Degree: Economics with Placement (Bachelors) - Bath University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Extended Project Qualification
Economics

“Studying for BSc Hons Economics, A level economics, maths and physics. Able to tutor GCSE/AS/A2 Economics, Maths and GCSE physics! ”

£20 /hr

Jacob F.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered:Maths, German+ 2 more

Maths
German
Further Mathematics
-Personal Statements-

“2nd year Maths undergrad with enhanced DBS (CRB) certification and experience teaching in secondary schools and at University. Fluent German speaker.”

About the author

£26 /hr

Daniel R.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
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

Split the following expression into partial fractions of the form A/(x-3) + B/(4x+2) : (19x-15)/(4x+2)(x-3)

If f'(x)=3x(x - 1), find f(x)

Solve the equation d/dx((x^3 + 3x^2)ln(x)) = 2x^2 + 5x, leaving your answer as an exact value of x. [6 marks]

Why does adding a constant to a function's input (as in f(x-a)) shift the plot of the function along the x-axis?

View A Level 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

mtw:mercury1:status:ok