Solve 4cos(2x )+ 2sin(2x) = 1 given -90° < x < 90°. Write 4cos(2x )+ 2sin(2x) in the form Rcos(2x - a), where R and a are constants.

Step 1: Recognise 4cos(2x )+ 2sin(2x) must be put into the form given in the question form i.e. Rcos(2x - a) as it contains only one trigonometric function

Step 2: Calculate values of R and a

a. Expand Rcos(2x - a) using the suitable compound angle formulae: cos(A - B) = cosAcosB + sinAsinB, in our case A = 2x and B = a
Hence Rcos(2x - a) = R[cos(a)cos(2x) + sin(a)sin(2x)]

b. By identifying Rcos(a) and Rsin(a) as effective constants and matching these to the original form we can obtain two trigonometric equations in R and a: Rcos(a) = 4 {eq.1} & Rsin(a) = 2 {eq.2}

c. Dividing {eq.1} by {eq.2} & canceling R gives tan(a) = 1/2, thus arctan(1/2) = a, yielding a = 26.6°

d. Squaring {eq.1} & {eq.2} and taking the sum gives: R^2(cos^2(a) + sin^2(a)) = 4^2 + 2^2 = 20
Noting that this step has yielded cos^2(a) + sin^2(a) and can be replaced by 1 using the standard trigonometric identity cos^2(y) + sin^2(y) = 1, this leads to the result R = sqrt(20)

Step 3: Solve the equation given for x

a. From the result in step 2, 4cos(2x )+ 2sin(2x) = 1 can be written as sqrt(20)*cos(2x - 26.6) = 1

b. Dividing by sqrt(20) and taking the inverse cosine yields: arccos[1/sqrt(20)] = 2x - 26.6 From your calculator arccos[1/sqrt(20)] yields 77.08° but recognising -77.08° (282.92°, -282.92° etc.) are also solutions crucial(important)
This can be seen by looking at a sketch of y = cos(t)

c. Solving for x in the interval given can be done by rearrangement
2x - 26.6° = 77.08°, -77.08° (282.92°, -282.92°)
Hence x = 51.84° -25.24°
Note that (282.92°, -282.92°) give values out of the range defined for x

Step 4: Check on your calculator that the calculated values of x solve the original equation 4cos(2x )+ 2sin(2x) = 1
Prannay K. A Level Chemistry tutor, GCSE Chemistry tutor, A Level Phy...

4 months ago

Answered by Prannay, an A Level Maths tutor with MyTutor

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


Michael T. A Level Maths tutor, GCSE Maths tutor, A Level Computing t...
View profile
£20 /hr

Michael T.

Degree: MMath (Masters) - Durham University

Subjects offered: Maths, Physics+ 1 more


“About Me: I am currently studying Mathematics at Durham University. Maths can sometimes be intimidating, but it's that which makes it so rewarding when it all starts to make sense. I have been teaching in one form or another since 13...”

PremiumTom W. GCSE Maths tutor, GCSE Chemistry tutor, GCSE Physics tutor, A ...
View profile
£26 /hr

Tom W.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Maths


“Third year Maths undergraduate, with experience teaching Maths from GCSE to first year.”

Pavit B. A Level Further Mathematics  tutor, GCSE Physics tutor, GCSE...
View profile
£20 /hr

Pavit B.

Degree: Physics (Masters) - Durham University

Subjects offered: Maths, Physics+ 1 more

Further Mathematics

“Who am I?Hi, I'm Pav. I'm currently a physics student at Durham University and there is very little I enjoy more than teaching. I have afasincation for the sciences and in my tutorials I hope to be able to get you not only interested...”

About the author

Prannay K. A Level Chemistry tutor, GCSE Chemistry tutor, A Level Phy...
View profile

Prannay K.

Currently unavailable: for new students

Degree: Engineering Science (Masters) - Oxford, Somerville College University

Subjects offered: Maths, Physics+ 3 more

Further Mathematics

“About Me: I am an engineering student at Oxford and have always enjoyed learning about maths and science. Hopefully I can bring that same enjoyment to you in my lessons. I am very patient and friendly. I have been teaching cricket sin...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

How do I integrate terms with sin^2(x) and cos^2(x) in them? For example integrate (1+sin(x))^2 with respect to x

How do I differentiate implicitly?

How do I know which method of integration to use?

Find the solutions to x^3+4x^2+x-5=1

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