Inspired by WishBoNe’s recent Bali escapade, Miss Loi decided to take a mini-break of her own and headed to a pristine beach somewhere in the region.

It was postcard-perfect day as she slipped into her sexy bikini, strutted to the waterfront and settled down with a glass of cocktail, oblivious to the lecherous pairs of eyes gawking from a distance.

Whatever postcard serenity was ruined by a sudden cry of “TSUNAMI!“, as Miss Loi looked up to see a distant wall of water hurtling towards her.

As the gawking eyes fled to higher ground, however, Miss Loi began to question the wisdom of abandoning her expensive cocktail mix and fleeing straight into the welcoming company of lecherous cheekopeks on higher ground, clad in a sexy bikini.

With her supernatural trigonometric powers, she managed to deduce the equation of the wave. But time is of the essence, and some questions still needed to be answered …

The wave f is defined, for 0^{o} ≤ x ≤ 180^{o}, by

f(x) = 2cos4x-1

Solve the equation f(x) = 0.

State the amplitude of f.

State the period of f.

State the maximum and minimum values of f.

Sketch the graph of y = f(x)

Given that the amplitude is in metres, and that the beach is at f(x) = 0, and that Miss Loi’s around 1.7m tall, and that her primary concern is not to ruin her expensive make-up that comes with UV protection, should she run in order to save her make-up life? If she stayed at the beach, how many ‘peaks’ of the wave would she need to survive, assuming the wave only lasted for the number of cycles within the range of 0^{o} ≤ x ≤ 180^{o}?

Please, there’s no time to lose before the waves come crashing down on the beach! Miss Loi needs you to really know your graphs of trigonometric functions well to answer this quickly!

That’s why she used the word “state” many times in the question, which means there’s no time for any workings for those parts!

P.S. This is a speed question i.e. it’s not about whether you know how to do but how fast you can do this, get your marks, and move on. Do not dwell > 5 mins on a question like this!

Disclaimer: This story is entirely fictitious. There was no tsunami (and they don’t behave according to that equation), no sexy bikini, and most importantly Miss Loi is definitely not 1.7m tall.

Ignorantsoup: Hope the 'powers that be' don't consider this as a case of 'misleading' blog content 😛

Peter: Sometimes ignorance may be a blessing in disguise. All the cheekopeks already enjoying the view from the hilltop and Miss Loi's still doing trigonometric calculation in the face of the tsunami!

eh its 2cos(4x-1) or 2cos(4x) -1 ? de ( ) is tremendously significant anyway i'l juz assume that

1) 0=2 cos(4x-1) 90 = 4x-1 x = 22.71(weird answer..)

but if i were to take it as 2cos(4x) -1 x would be 15

For such O-Level A-Maths questions, the equation is usually in the form of acosbx + c, so you should take this as 2 cos(4x)-1.

Anyway ... YOU GOT PWNED BY THE TSUNAMI!

What you did was f(x) = 2cos(4x) -1 = 0 ⇒ cos(4x) = 1/2

And using your super-duper calculator, you got 4x = 60^{o} and then you divide by 4 to find x?

NOT SO SIMPLE DUDE!

The function lies within the range 0^{o} ≤ x ≤ 180^{o}. So you'll need to find ALL the values of x that satisfy the condition f(x) = 0 within this range. That 60^{o} is just your basic angle (= α).

Remember this classic All Science Teachers are Crazy diagram?

Since cos(4x) is POSITIVE, α falls within quadrants A and C. Which means:

4x = 60^{o} (in quadrant A), 360^{o}-60^{o} = 300^{o} (in quadrant C). ⇒ 4x = 60^{o}, 300^{o}

N.B. For those who are lost please start referring to your textbook now.

But we're not done yet, because we're dealing with 4x here, which means we need to go a total of 4 rounds round the 360^{o} circuit, and we've only completed 1 round. But since we're only interested in 0^{o} ≤ x ≤ 180^{o}, we only need a total of 2 rounds in the circuit.

Please familiarize yourself with this process. It looks involved but it's really straightforward and there's not much trickery. So don't lose marks here!

2) I'll assime its 2cos(4x) -1 if not i dun see how it's possible to do

-1< cos4xf(x)>-3

YOU GOT PWNED BY THE TSUNAMI AGAIN!

This portion deals with the a portion of your acosbx + c graph.

Many students got tricked here. Remember the amplitude of the trigonometric function does not change even when the graph is shifted up/down! So since the amplitude should still be a i.e. 2!

3)period = 180 / 4 = 45 (i think..still not quite sure what period is, lazy to check textbook)

YOU GOT PWNED BY THE TSUNAMI AGAIN!

This portion deals with the b portion of your acosbx + c graph.

The period is defined as the interval over which the curve repeats itself. A cosx graph repeats itself over 360^{o}, so for cos(4x) the graph repeats itself over 360^{o}/4 = 90^{o}

4) max =1 , min = -3

OK, the Tsunami didn't get you this time.

This portion deals with the c portion of your acosbx + c graph.

Since c = -1, the entire graph shifts downwards by 1. Correspondingly the max point (2) and min point (-2) gets shifted down by 1. Hence you're right!

Kiroii, looks like you've been pretty ravaged by the Tsunami! Please take a look at Miss Loi's comments on your answers as this is a pretty important topic for your upcoming exam.

FYI parts 1 and 5 (those without the word state, obviously) carry the most marks, so you'll need to spend the majority portion of your 5 min on these. Parts 2-4 are pretty much *ping* *pong* *poom* i.e. done under a minute!

While we're at this, Miss Loi might as well present the sketched graph for part 5:

Note that all answers from parts 1-4 were used and indicated in the sketched graph. And pay attention to the fact that the graph is only sketched for 0 ≤ x ≤ 180, as stated in your question - so always read your question carefully.

So, assuming Miss Loi is at Ground Zero (y = 0) do you think she should run? 🙂

wow..de first question is engineered to kill..that all i gotta say..totally 4got about about your approach..it seemed to easy>.< still waters run deep i guess

At least the water's still and there's no hidden whirlpool 'engineered to kill you'.

This is something basic and everyone should know this. Moreover you won't be able to sketch the graph properly in part 5 without the answers from part 1.

But to be honest Miss Loi did had a premonition that some students will do a simple cos^{-1}x and call it a day. She actually wanted to put up the SATC diagram within the blog post, but she decided against it as one glance will be enough to hint to you what you should do. But always remember that in your exam, there's no such luxury and nobody will be there to prompt you.

So here's the same old naggy tagline: Always know your approach before you jump into the question!

* Please refer to the relevant Ten-Year Series for the questions of these suggested solutions.

About Miss Loi

Miss Loi is a full-time private tutor in Singapore specializing in O-Level Maths tuition. Her life’s calling is to eradicate the terrifying LMBFH Syndrome off the face of this planet. For over years she has been a savior to countless students … [read more]

Gratitudes

Her lessons are very well-structured, and she has a strange ability to make you remember complex formulae within minutes … [read more]

## 13 Comments

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i just love the way you phrase your questions. 🙂

曜

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Since I suck at maths....

I'm just gonna RUN!!!

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Ignorantsoup:Hope the 'powers that be' don't consider this as a case of 'misleading' blog content 😛Peter:Sometimes ignorance may be a blessing in disguise. All thecheekopeksalready enjoying the view from the hilltop and Miss Loi's still doing trigonometric calculation in the face of the tsunami!曜

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eh its 2cos(4x-1) or 2cos(4x) -1 ? de ( ) is tremendously significant anyway i'l juz assume that

1) 0=2 cos(4x-1)

90 = 4x-1

x = 22.71(weird answer..)

but if i were to take it as 2cos(4x) -1

x would be 15

For such O-Level A-Maths questions, the equation is usually in the form of

acosbx+c, so you should take this as 2 cos(4x)-1.Anyway ... YOU GOT PWNED BY THE TSUNAMI!

What you did was f(

x) = 2cos(4x) -1 = 0⇒ cos(4

x) = 1/2And using your super-duper calculator, you got 4

x= 60^{o}and then you divide by 4 to findx?NOT SO SIMPLE DUDE!

The function lies within the range 0

^{o}≤x≤ 180^{o}. So you'll need to find ALL the values ofxthat satisfy the condition f(x) = 0 within this range. That 60^{o}is just yourbasic angle(=α).Remember this classic

AllScienceTeachers areCrazy diagram?Since cos(4

x) is POSITIVE,αfalls within quadrantsAandC. Which means:4

x= 60^{o}(in quadrant A), 360^{o}-60^{o}= 300^{o}(in quadrant C).⇒ 4

x= 60^{o}, 300^{o}N.B. For those who are lost please start referring to your textbook now.

But we're not done yet, because we're dealing with 4

xhere, which means we need to go a total of 4 rounds round the 360^{o}circuit, and we've only completed 1 round. But since we're only interested in 0^{o}≤x≤ 180^{o}, we only need a total of2 roundsin the circuit.So to go one more round,

4

x= 60^{o}(round 1), 300^{o}(round 1), 60^{o}+360^{o}(round 2), 300^{o}+360^{o}(round 2)⇒ 4

x= 60^{o}, 300^{o}, 420^{o}, 660^{o}⇒

x= 15^{o}, 75^{o}, 105^{o}, 165^{o}Please familiarize yourself with this process. It looks involved but it's really straightforward and there's not much trickery. So don't lose marks here!

2) I'll assime its 2cos(4x) -1 if not i dun see how it's possible to do

-1< cos4xf(x)>-3

YOU GOT PWNED BY THE TSUNAMI AGAIN!

This portion deals with the

aportion of youracosbx+cgraph.Many students got tricked here. Remember the

amplitude of the trigonometric function does not change even when the graph is shifted up/down! So since the amplitude should still beai.e. 2!3)period = 180 / 4

= 45 (i think..still not quite sure what period is, lazy to check textbook)

YOU GOT PWNED BY THE TSUNAMI AGAIN!

This portion deals with the

bportion of youracosbx+cgraph.The period is defined as the interval over which the curve repeats itself. A cos

xgraph repeats itself over 360^{o}, so for cos(4x) the graph repeats itself over 360^{o}/4 = 90^{o}4) max =1 , min = -3

OK, the Tsunami didn't get you this time.

This portion deals with the

cportion of youracosbx+cgraph.Since

c= -1, the entire graph shiftsdownwardsby 1. Correspondingly the max point (2) and min point (-2) gets shifted down by 1. Hence you're right!曜

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Kiroii, looks like you've been pretty ravaged by the Tsunami! Please take a look at Miss Loi's comments on your answers as this is a pretty important topic for your upcoming exam.

FYI parts 1 and 5 (those without the word

state, obviously) carry the most marks, so you'll need to spend the majority portion of your 5 min on these. Parts 2-4 are pretty much *ping* *pong* *poom* i.e. done under a minute!While we're at this, Miss Loi might as well present the sketched graph for part 5:

Note that

allanswers from parts 1-4 were used and indicated in the sketched graph. And pay attention to the fact that the graph is only sketched for 0 ≤x≤ 180, as stated in your question - so always read your question carefully.So, assuming Miss Loi is at Ground Zero (

y= 0) do you think she should run? 🙂曜

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If missloi is at ground zero, I'd give her a life jacket 🙂

THEN I'll run!

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What a 'hero'! 😕

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wow..de first question is engineered to kill..that all i gotta say..totally 4got about about your approach..it seemed to easy>.< still waters run deep i guess

曜

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At least the water's still and there's no hidden whirlpool 'engineered to kill you'.

This is something basic and everyone should know this. Moreover you won't be able to sketch the graph properly in part 5 without the answers from part 1.

But to be honest Miss Loi did had a premonition that some students will do a simple cos

^{-1}xand call it a day. She actually wanted to put up the SATC diagram within the blog post, but she decided against it as one glance will be enough to hint to you what you should do. But always remember that in your exam, there's no such luxury and nobody will be there to prompt you.So here's the same old naggy tagline: Always know your approach before you jump into the question!

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T_T O in 6 days..... stressssssss

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ya 6days...till i get to whack the paper question by question *charges my power further beyond cream de la cream power levels"

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123 & Kiroii: Gambate!!! 6 (actually 7) more days to burning your A-Maths textbooks!曜

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noooooooooooooooo...... I haven learn trigo!! if i'm there, i'll most likely die XP