Like Honda’s Type R variant of cars that appeal to *Ah Beng* racer boys who like to ~~bully~~ weave in front of Miss Loi’s demure little car on the expressways, Trigonometry has its own “Type R” model in the form of the **R-Formula** that was unveiled in this year’s New AMaths Syllabus.

As such students are warned to listen out for the roar of the **R-Formula** whenever you encounter a trigonometric question:

with an expression that contains a

**constant**, like this:Solve for 0° <`x`< 360°,

2 cot`x`= 3 + 2 cosec`x`or where you’re asked to find an expression’s max/min value, like in this rather

*cheong hei*(long-winded) question:The diagram shows two right-angled triangles with their right angles at

`B`and`N`. The sides`OB`and`AB`are of length 6 cm and 2 cm respectively. The lines`OB`is inclined at angle`θ`to`ON`. The line`AM`is perpendicular to`ON`.- Express
`OM`in the form`OM`=`R`cos (`θ`+`α`) and calculate the values of`R`and`α`. - Find the value of
`θ`for which`OM`= 5 cm. - In the diagram, state the line with a length of
`R`cm and the angle with a value`α`. - Express
`AM`in terms of`R`and (`θ`+`α`) and show that the area of triangle`OAM`is 10 sin 2(`θ`+`α`). - Given that
`θ`can vary, find the maximum value of the area of triangle`OAM`and the corresponding value of`θ`when this occurs.

- Express

To handle the new **R-Formula** questions, students need to recall the following expressions, as **they will not be appearing on your formula sheet during your actual exam**:

### The R-Formula

`a`cos`θ`±`b`sin`θ`=`R`cos (`θ`∓`α`) – look out for that inverted plus/minus!`a`sin`θ`±`b`cos`θ`=`R`sin (`θ`±`α`)

where and

and `a` > 0, `b` > 0, `α` is *acute*

#### Maximum/Minimum Values of R-Formula Expressions

- The maximum value is
`R`which occurs when cos/sin (`θ`±`α`) = 1 - the minimum value –
`R`which occurs when cos/sin (`θ`±`α`) = -1

#### Deriving The R-Formula

Do note that the R-Formula highlighted above is NOT a given, especially when the relevant part of the question carries a **high weightage of marks** e.g.

`θ`– 3 sin

`θ`in the form of

`R`cos (

`θ`+

`α`) [6 Marks]

There’s a bit of debate on this but don’t think you can get away with all 6 marks here by simply substituting the values into the R-Formula! So to be a little *kiasu*, it’s advisable to *derive* the whole thing as per the following steps:

4 cos `θ` – 3 sin `θ`

= `R` cos (`θ`+`α`)

= `R` [cos`θ`cos`α`-sin`θ`sin`α`] (Addition Formula)

= (`R` cos`α`)cos`θ`-(`R` sin`α`)sin`θ`

Comparing coefficients between LHS & RHS,

`R` cos`α` = 4 —– (1)

`R` sin`α` = 3 —– (2)

(2)/(1): tan `α` = 3/4 ⇒ `α` = 36.9°

(1)^{2}+(2)^{2}:

`R`^{2}cos^{2}`α`+`R`^{2}sin^{2}`α` = 4^{2}+3^{2}

`R`^{2}[cos^{2}`α`+sin^{2}`α`] = 25 ⇒ `R` = 5

∴ 4 cos `θ` – 3 sin `θ` = 5 cos(`θ`+36.9°)

#### Important Notes

To reiterate what was stated above, `a` and `b` MUST be positive. DON’T do this:

(-3) cos `θ` + (-2) sin `θ` = 3 **→ WRONG!**

Instead, move the terms around to ensure that the first term is always positive:

3 cos `θ` + 2 sin `θ` = -3 **→ CORRECT! YAY!**

Please, please, please, please, please for the *addition* cases (i.e. `a` cos `θ` + `b` sin `θ` or `a` sin `θ` + `b` cos `θ`) you can simply **re-arrange** the sine and cosine terms depending on how you wish to express the R function in your question.

E.g.

3 cos `θ` + 2 sin `θ` → if you require `R` cos (`θ` – `α`)

2 sin `θ` + 3 cos `θ` → if you require `R` sin (`θ` + `α`)

Hailed as the pinnacle of O-Level Trigonometry, a typical **R-Formula** question may also require your skills on lesser trigonometry variants such as the *Addition Formulae*, or the *Double Angle Formulae*, or the *Factor Formulae*, together with your knowledge of basic angles (e.g. in question 1 above when you’re asked to find *all* angles within the range of *blah blah blah*).

But once you tame it, you’ll be rewarded with a straightforward ride and zoom away with plenty of marks.

So are you ready to test-drive your **R-Formula** with the two questions above? Miss Loi doesn’t expect you to crash this in your O-Level in two weeks’ time!

*Zooms away for Jφss Sticks Sessions (Type-R variant)*

N.B. Once again, Miss Loi does NOT advocate nor encourage speeding or any form of street racing in Singapore.

## 24 Comments

曜

日

I'm trying this for fun =) (for lbmfh practice actually)

Miss Loi is now marking your answer in a LMBFH manner too 😛 ...

1)Is there only 1 answer? My only answer is 247.4

Yes you're right there's only one answer! To elaborate, we have

2 cot

x=3+ 2 cosecxSo seeing that lonely constant 3 in the equation should signal to you that you're going to start the

Trigonometry R-Formulaengine *VRROOOOOM*Expanding ...

(Note: we put the [2 cos

x] as the first term so as to keepa(in theR-Formula) positive)To express the LHS above in

Rcos (x+α) form, we haveR= √(2^{2}+3^{2}) = √13tan

α= 3/2 ⇒α= 56.3°(Note: Both

a&bhas to be > 0 when you're obtainingα! DON'T do a tanα= -3/2!)So we now have:

√13 cos (

x+ 56.3°) = 2⇒ cos (

x+ 56.3°) = 2/√13This is where many students get pwned and forget that

AllScienceTeachers areCrazy (ASTC), as the question asks forALLvalues ofxin the range of 0° < x < 360°.So doing a cos

^{-1}2/√13 in your calculator will only yield abasic angleof (x+ 56.3°) = 56.3°. But since 2/√13 is positive, we see from the ASTC diagram below that:⇒

x+ 56.3° = 56.3°, 303.7° (1st and 4th quadrant)⇒

x= 0°, 247.4°But we shall take only

x= 247.4° since 0° is not in the range of 0° < x < 360°!2)(i)√40cos(θ+18.43)

Once again, to elaborate on your answer (next time show workings okay? Tsk tsk.) ...

Typically in a question like this, we first need to express

OMin terms of sinθand cosθBEFORE we can use theR-Formula. From the following diagram,... you should be able to see that ∠

MAB=θ, and sinceOM=ON-MN⇒

OM= 6 cosθ- 2 sinθ(using your TOA CAH SOH)

Using the

R-Formula,OM=Rcos (θ+ α) ----- (1)R= √(6^{2}+2^{2}) = √40α= tan^{-1}2/6 = 18.435°⇒

OM= √40 cos (θ+ 18.435°)(ii)56.2

Think you've made a careless mistake here but Miss Loi won't know in her lifetime the cause of it since your workings are absent!

In anycase, simply equate

OMto 5cm and find the value ofθwhen this occurs:√40 cos (

θ+ 18.435°) = 5cos (

θ+ 18.4°) = 5/√40(

θ+ 18.435°) = 37.761°, 322.239°Once again don't forget to A-S-T-C thingy BUT BUT BUT in this case we're sure from the diagram that

θis acute so we can safely discard the 322.239° and conclude that:θ= 37.761° - 18.435° = 19.3° whenOM= 5cm(iii)AO, angle AOB

This part requires a bit of 阴阳眼 (The Third Eye) which

Anoneemershas demonstrated that he obviously possesses one 😉You should be able to 'see' this from the diagram above 😉

(iv)AM=Rsin(θ + α)=√40sin(θ+18.43)

Area of OAM

=(0.5)[√40sin(θ+18.43)][√40cos(θ+18.43)

=20sin(θ+18.43)cos(θ+18.43)

=10sin 2(θ+18.43)

WOAH you have workings now! Yes, from the diagram in Part iii., you can easily see that

AM=Rsin (θ+α) ----- (2) simple right???But if you read carefully again, Part iv. only requires you to

expressin terms ofRand (θ+α) (vs Part i. which required you tocalculatethe values ofRandα). So strictly speaking you shouldn't replaceRand α with their actual values in your answer at this point.Now area of Δ

OAM= 0.5 xOMxAM. Using the expressions in (1) and (2),Area of Δ

OAM= 0.5 x

Rcos (θ+ α) xRsin (θ+α)= (0.5)

R^{2}(0.5) 2 cos (θ+α) sin (θ+α)(when you see a sine and cosine term with the same angle together - you should instinctively think of the

Double Angle Formulafor sine!)Sub in

R= √40 obtained from Part i.= (√40

^{2}/4) sin 2(θ+α)= 10 sin 2(

θ+α) YAY!(v)10

Marks are usually served to you on the platter in the Maximum/Minimum parts of R-Formula questions so DON'T EVER 对不起你自己 and skip this part!

From the notes above: The maximum value is

Rwhich occurs when cos/sin (θ±α) = 1.So in our case the maximum area is simple 10 cm

^{2}(the coefficient of our expression in Part iv) and this occurs when sin 2(θ+α) = 1⇒ sin 2(

θ+ 18.435°) = 1⇒ 2(

θ+ 18.435°) = 90°⇒

θ= 26.6° (sinceθmust be acute)By the way how do u type all those square roots and teeter symbol i had to copy and paste =.=

It's my 1st time making

曜

日

Hey

Anoneemers, Miss Loi has commented on your test drive of the Trigo R-Formula questions.Regarding the symbols, Miss Loi was taught from the start to reference this table. But after awhile it became pretty intuitive e.g. for α you type α, for β you type β etc.

But having said that, copy and pasting still seem to be the quickest!

曜

日

What? This thing is in O level A maths nowadays? We're moving closer to this day...

Used to take a long time remembering this when I did A levels, and until now still cannot figure out the logic behind all those steps used to derive the Rsin and Rcos.

曜

日

mathslover:As far as Miss Loi can recall (please correct if she's wrong though), the R-formula was only removed from the A-Math syllabus between 2002-2007, as she could still remember herself utilizing the R-Formula diligently when she was still akawaiiteeny Sec Four school girl.So her

kaypohguess is you're around 18-23 yrs-old? Oh wait ... but you know how to do Plane Geometry questions!The plot thickens ...

曜

日

Miss Loi: Actually, my O level year not within the range.~~Its the extraordinary year where matrices and vectors are out.~~Guess either u remember wrongly, or i was sleeping in A maths class..

Plane Geometry is new? I kinda faintly remembered doing something called Circular Measure which include tangent/chord/segment/sector...

No plot

lahMiss Loi, you~~watch too much HK drama~~put yourkawaiinotes below the prawns so big, anyone can do the question le. 🙂曜

日

Help! I can't get the answer for this question, 4sin3x + 3cos3x=√3.

Tried doing the question for 2-3 times but still can't get the answer.

I don't understand the minimum and maximum thing 🙁 🙁

曜

日

Hello

Rose, yours is really a classictrigonometric equation question.asinθ±bcosθ=cFirst understand that the

R-Formula part i.e. converting the LHS into aR-Formula expression, is straightforward:Sub

a=4,b=3 for *Rsin (θ±α) (do check with your Cher if you need to derive theR-Formula usingAddition Formula, especially if the part carriesa lot of marks),R= √(a^{2}+b^{2}) = √(4^{2}+3^{2}) = 5α= tan^{−1}= tan^{−1}≈ 36.870°Thus your LHS: 4 sin 3

x+ 3 cos 3xcan be expressed as:5 sin (3x+ 36.870°)*Note:Since both terms on LHS are positive, and assuming the question didn't explicitly state which form to express in, you may also express it in the form ofRcos(θ−α) but in this case youra=3 andb=4!To address your confusion of the "minimum and maximum thing", as Miss Loi has no further detail of your question, let's assume that we're looking for the max/min values of 5 sin (3

x+ 36.870°).Consider the typical graph of a sine function

y=asinθwhich you learnt in Sec 3:Looking at your

R-Formula expression of 5 sin (3x+ 36.870°), can you see that it's also a sine graph when you leta=5 andθ= 3x+36.870°? Hence from the diagram above, the max value is 5 and the min value is −5.Now to solve for

xin your equation (the part where many students get KIA because they often forget about their basic angles):We have:

5 sin (3

x+ 36.870°) = √3sin (3

x+ 36.870°) =Basic angle

β= sin^{−1}= 20.268°Assuming your question asks for all values of

xin the range 0° ≤x≤ 360°,Since sin (3

x+ 36.870°) is positive,⇒ 3

x+ 36.870°= 20.268°, 159.732°, 380.268°, 519.732°, 740.268°, 879.732°

Note that since it's 3

xin the expression, we need to go3 roundsround the SATC diagram i.e.β+ 0°,β+ 360°,β+ 720°3

x= −16.602°, 122.862°, 343.398°, 482.862°, 703.398°, 842.862°∴x≈ 354.5°, 50.0°, 114.5°, 161.0°, 234.5°, 281.0°Does the answer tally with yours?

曜

日

Thank you!

Erm... Can you teach me how to find the max and min values of (5sinx - 12cosx)²?

曜

日

Hello again

Roseand a interesting but nonetheless a pretty straightforward question you have!To find the max/min values of (5 sin

x− 12 cosx)², you may first express the expression within the brackets using theR-Formula i.e:(5 sin

x− 12 cosx)² = [Rsin (x−α)]²As discussed earlier throughout this blog post,

we know

R= √(5²+12²) = 13⇒ (5 sin

x− 12 cosx)² = [13 sin (x−α)]²Since max/min value of sin (

x−α) = ±1, we can deduce that the max/min value of 5 sinx− 12 cosx= ±13BUTwe're dealing with aSQUAREDexpression (5 sinx− 12 cosx)² soMax value = 13² = 169

Min value = 0 (since the square of a −ve term is always +ve)

Happy New Year!

曜

日

I am weak in trigonometry, and realise that it stands a huge amount of marks in paper 2. I'm a Sec 4 student and MYE is coming :S

曜

日

@AMath Student: Yes for e.g. in the TYS's analysis of the 2008 GCE 'O' Level AMaths paper, Trigonometry (Trigo functions, identities & equations, R-formula etc.) took up a whooping 39 marks (21.6%) of the total marks across Papers 1 and 2.

This is second only to calculus i.e. Differentiation & Integration (45 marks/25%) which incidentally also contains trigonometric functions 😉

Good luck for your Mids! Take this as a good practice opportunity for almost 90% of your O Level topics - though Miss Loi can't wait for them to end at the moment 😕

曜

日

i m weak in math s ive an exam on 1 june please guide me wat can id o to get a seat in bitsat

曜

日

Miss Loi is teta = 90°- alpha in the first triangle?

曜

日

sorry I was referring to the second diagram I didnot see the first one?

曜

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@Jukie Primestein: Are you referring to this diagram?

If so, you'll find in Δ

OABfrom the diagram above thatθ< 90° −α曜

日

Can you please send me a few more variation of the roar R-formula. Based on a few times I encountered it it is not always clear if you are to find a maximum or minimum! Oh another question I have not seen any proof to convince me than cosine of a negative number is positive, but please don't get me wrong I know it is correct however a proof maybe through a diagram would really do the trick!

曜

日

which trig identity do you refer to as the factor formula?

曜

日

@Jukie Primestein: If Miss Loi gets your question right ... the Factor Formulae refer to

Are they called the same thing over at Reggae Jamaica? 😉

曜

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why cannot r be negative 5 in the exaple given ?

曜

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Hi Miss Loi,

I love your site and your sense of humour, definitely very localised.

Anyway, a quick question to your type-R question 2, the part on maximum area of the triangle.

Should we apply the maximum value of R at the length stage? meaning to AM and OM individually instead of applying to the area straghit?

The maximum value of AM and OM exist when cos (θ + α) =1, Hence, maximum value for AM and OM should be √40. Hence, 1/2 times AM times OM should give us 20cm sq.?

I'm confused. Pls clarify if you can? Thanks! =]

曜

日

@Ray: Hello and welcome to Jφss Sticks, Ray!

From parts 2(i) & 2(iii) of the workings here,

OM= √40 cos (θ+18.43°)AM= √40 sin (θ+18.43°)This means that both the lengths of

OMandAMare dependent on the same angleθ, which, looking at the equations above, means that both cannot be maximum at the same value ofθi.e. When

OMis maximized when cos (θ+α) = 1,AMis actuallyminimizedat this same value ofθsince sin (θ+α) = 0.So the key here is we need an intermediate value of

θthat will maximize the area instead of the individual lengths hence we have to obtain theONER-formula for the area using the double-angle formula etc.曜

日

Hi. I was wondering if you can help me with these questions.

8 cos x - 15 sin x = 17 cos (x + 1.08) -------- (R formula)

1. Find the minimum and maximum value of 20 - 8 cos x + 15 sin x and the corresponding x values for 0 ≤ x ≤ 2π.

2. Solve the equation |8 cos x - 15 sin x| = 6 for 0 ≤ x ≤ 2π.

Thanks in advance!

曜

日

@Agnes: Look ma! A math question in the comments section! ^^

So you have the

R-formulaAnd from the notes above, the maximum/minimum value of a

R-formula is ±RsoAt this juncture, there'll be a insatiable urge to flip the centre term and make the careless mistake of flipping the inequality signs, +20 on both sides to get what you want i.e.

(✘)

Not only does the above don't make any mathematical sense (3 ≥

x≥ 37!), one has to always consider thenature of the graphyou're dealing with when it comes to inequalities. From your Trigonometric Equations & Graphs, you know that the max/min values of aAcosxcurve is still ±A, even when you flip it around for −Acosθ, as shown:So you may now 安安心心 proceed in a as follows:

∴ Maximum value of

37occurs whenand minimum value of

3occurs whenNote:

x= −1.08 rejected ∵ 0 ≤x≤ 2π曜

日

@Agnes: For the second part, we simply expand the modulus function as follows

Using the SATC quadrants, we have for cos

^{−1}(6/17):and for cos

^{−1}(−6/17)