Trig Identities #3 Periodicity


You can hopefully see from the diagram above, that if we add 2\pi to our
angle \theta , we would come ‘full circle’ (ba-dum tish) and be back where we started from (think of it as adding 360 degrees if it makes you happier).


(\cos(\theta \pm 2\pi), \sin(\theta \pm 2\pi)) = (\cos(\theta), sin(\theta))

\cos(\theta) = \cos(\theta \pm 2\pi)

\sin(\theta) = \sin(\theta \pm 2\pi)

And since csc and sec are basically just utilisations of sin and cos, then they have the same property:

\csc(\theta) = \csc(\theta \pm 2\pi)

\sec(\theta) = \sec(\theta \pm 2\pi)

Now  \tan \theta has a periodic cycle half the size of \sin \theta , which you can perhaps visualise easily by studying their respective plots side by side:


and so :

\tan(\theta) = \tan(\theta \pm \pi)

\cot(\theta) = \cot(\theta \pm \pi)


Trig Identities #2 Pythagorean Identities

trig 2

Above is a quick refresher on each trig function. You only really need sin, cos, tan – but for convenience:

tan(\theta ) = \frac{sin(\theta )}{cos(\theta )}

(think SOHCAHTOA on the unit circle)

csc(\theta ) = \frac{1}{sin(\theta )}

sec(\theta ) = \frac{1}{cos(\theta )}

cot(\theta ) = \frac{1}{tan(\theta )} = \frac{cos(\theta )}{sin(\theta )}


Recall the formula of a circle is


and if our x and y coordinates (cos(\theta), sin(\theta)) lie on the circle (which they do) then:

cos^2{\theta } + sin^2{\theta } = 1

Take this identity and divide both sides by sin^2{\theta } :

\frac{cos^2(\theta )}{sin^2(\theta)} + 1 = \frac{1}{sin^2(\theta)}


We could instead have divded both sides by cos^2{\theta}  :

1+ \frac{sin^2(\theta )}{cos^2(\theta)} = \frac{1}{cos^2(\theta)}

1 + tan^2(\theta) = sec^2(\theta)


I’m a mathematician. Even writing it seems a little odd. I never identified as one until people started referring to me as such a few years ago. I’ve studied and/or taught mathematics pretty much my whole life,  which you’d think would automatically qualify – but calling yourself a mathematician is hard to do. Why? Because the very idea of what a mathematician is has become a polluted mess.

“A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.”

That word extensive…

It seems the title of ‘mathematician’ is reserved by many people for those at a mystical peak of mathematical powers – they solve problems like super humans in split seconds, or gain insight into approaches to seemingly impregnable problems at first glance. Their capes wrestle against the wind as they flawlessly answer a random question posed to them on the spot. Calculator Man and Wolfram Girl, nothing can stop them! You cannot catch them out, and they’ll always pick you up on your errors. “Oh but you’re assuming we’re not working in ring theory then?”. Of course they know about ring theory. All mathematicians know EVERYTHING about ALL elements of maths. You can give them an exam paper from Cambridge on an obscure module about astrophysics and they’ll soon crack it. It’s just maths right? You’re a mathematician, you can do it. We’re all our own accountants, and we could do an actuary’s job for them if we wanted, after all, it’s what we do. Number stuff. All of it.

But what if that perception is a teensy bit unrealistic? What if we’re only good at some bits of maths? What if we’ve never even heard of some other bits? What if we just enjoy doing maths? Does that make us weak? A weak mathematician, or not even a mathematician at all, just an enthusiast at best. A novice. Does this apply to other areas? Let’s see. Is a runner an athlete? They’re not high jumpers though… Do all doctors do brain surgery? At what point do I go from playing the guitar to becoming a musician? Could it be that we have slightly unrealistic perceptions of the supernatural abilities of ‘a mathematician’? We seem to be a special case, and I’m pretty sure I know where it comes from.

Think for a minute about what gets emphasised in school – from year 1 to year 13. Answers and speed, answers and speed. Timed tests, tricks and ticks. ‘This’ gets you the answer, doing ‘this’ gets it quicker. This is right, so well done – but this is wrong, so this is bad. Too slow, time’s up, so it’s not good enough. It’s good that you thought about it, but you didn’t get there, sorry pal. I’m going to point at you and you need to answer this question immediately. I’m the person at the front, I’m your teacher, I get this stuff right all the time (because I have the answers).

Well I say it’s time to reject all that. If you do maths, you’re a mathematician. This elite superpower we’ve invented doesn’t exist, so let’s stop being silly. I am not part of that. I’m crap at some things in maths, I don’t understand some things in maths, and I don’t even know about all sorts of things in maths. I make mistakes, I can do some things quickly and some things slowly, and I don’t care. Sometimes I can’t solve a problem, sometimes I can. Some days I can do everything right, and some days I seem to do everything wrong. It doesn’t matter. I’m a mathematician.

My Favourite Number

I like to ask trainees what their favourite number is in mathematics, on their first day. This inevitably goes the same way every year.

T1: “seven”

Me: “why?’

T1: “i don’t know”

T2: “three – because i have three children”

T3: “nine – because i live at 9 blah blah close”

I have a purpose to this task – two in fact. The first is that this is a question children ask maths teachers all the time. They assume we all sit studying numbers all day (a few of us genuinely do!) and that we have determined which are the ‘best’ numbers, and (perhaps paradoxically more interestingly) which numbers are less interesting. The whole idea of interesting and less interesting numbers is encapsulated nicely by the interesting number paradox which states

“all natural numbers are interesting. The “proof” is by contradiction: if there exists a non-empty set of uninteresting numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction.”

There’s a twitter account somewhere that capitalises on this idea, but I can’t remember it. If you know it let me know in the comments.

Anyway I’m digressing, the SECOND purpose of the exercise is to supply trainee maths teachers with cool answers so that they’re prepared for when they get asked the question “What is your favourite number?” or “which number is the best?”. If we give them some single digit thing and sentimental reasons, it’s a little unsatisfying for a curious mind isn’t it? I think so. So if you’re reading this and in desperate need for a more interesting number for those rainy days, here are a few of my favourites:

2520 – The lowest common multiple of 1,2,3,4,5,6,7,8,9,10

103003 – known delightfully as a millinillion

Graham’s Number

1729 – Ramanjuan’s number (the smallest number that can be expressed as the sum of two cubes in two different ways – 1729 = 13 + 123 = 93 + 103

18446744073709551615 – the solution to the ‘rice and chessboard’ problem (fair enough, this might be a little hard to recall on the spot… but practice makes perfect!). This problem dates back to around 1000 AD, i’ll blog about it one day. It basically involves putting one grain of rice on square one, then doubling the amount of rice for each subsequent square, and finding the sum.

44488 – the first of five consecutive happy numbers. A happy number is defined by the following process:

“Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1.Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).”


  • 6251 = 625
  • 6252 = 390625
  • 6253 = 244140625
  • 6254 = 152587890625
  • 6255 = 95367431640625
  • 6256 = 59604644775390625
  • 6257 = 37252902984619140625

Let me know more interesting numbers / your favourite number (and why)  in the comments. 🙂

Teachers are amazing.

For the last few days I’ve been thinking about the art of teaching – in part because it’s my job to train people to teach and I have a new set of trainees, but also because my recent(ish) trip to Japan was a bit of a revelation in so much as discovering how highly regarded teaching as an art actually is.

There’s a lot of talk about mastery of mathematics these days, but mastery of teaching is, currently, of greater interest to me.

I wonder how often we stop and think about just how intellectually demanding teaching is, and whether more emphasis on this might in fact encourage more into the profession. It’s not just about dealing with children – you have to be incredibly smart to do it well. In Japan, there is a real awareness of the need for constant refinement and improvement, and a collective kind of hive mindset about how to achieve this. Japanese teachers are acutely aware that there is no such thing as a perfect lesson, or a perfect teacher, and that it is a profession where you can only strive to get better and better, and continue your own learning and tailoring of your craft. I don’t know enough about Japanese culture to comment on whether this applies to all professions, but I am at least aware that the same concept applies to the world of martial arts. I’m not trying to compare the skill of transmitting knowledge about quadratic equations to that of taking down an assailant with Chuck Norr-ease, but in more general terms, the ‘forever learning’ concept applies to both. There’s something deeply satisfying in being part of a profession where you can constantly get better and refine your skills. It’s not something I always appreciate, but take a counter example: Imagine a job where you can master everything required of you in a matter of months, or even weeks. That feels awful in comparison when I think about it. Imagine sitting contemplating the idea that you have reached the point where you simply have nowhere further forward to go intellectually, but you have years of your working career ahead of you. Often people who change from other professions into teaching cite motives such as ‘I want to make a difference’ or ‘every day is different’. I guess the reasoning behind these statements isn’t too far removed from the desire to be intellectually stimulated.

It’s only when trainees have been teaching for a while that they start to fully appreciate just how many micro-decisions are involved in teaching. Behaviour management alone is a careful tightrope of reading people, anticipating their responses before (!), during and after they’ve made them, and subtly manipulating them to a place where their likely actions are the most predictable and controllable. It’s hugely complicated, but as it becomes part of daily working life, we probably don’t step back and notice how amazing it is that we’re even able to do it. All that before we even consider the difficulties of imparting knowledge and abstract concepts onto young impressionable minds. Take a moment to reflect on the complexity of the skills you possess. You’re fucking brilliant.


Problem Solving #4

I stumbled across this question in a book yesterday:



I know the problem, and in fact have solved it a few times before, but upon looking at it I couldn’t quite remember how I solved it, so I thought I’d have another stab at it. It didn’t go well…!


I knew I needed something like this diagram, because there’s just nothing else I can think of doing with the tiny amount of information I’m given.


More specifically, I just need this half (I think). But this is where I got into a pickle. I stared at the above diagram for a little while, convinced it was what I needed, but completely lost at how it helps me. Sure I have one side, and what I need is part of the triangle, but surely I need *more* information??

Now as I said, I have solved this problem a few times, and in fact, the first time it was shown to me I solved it in about a minute or two, but for whatever reason I couldn’t for the life of me get past this point yesterday. The reason why I couldn’t progress was simple – I wasn’t following my own advice on solving problems. Specifically, I’d recommend to students just finding as many things as you can until you find a way in. So after staring at that little triangle for far too long, I decided to just use Pythagoras anyway, despite “knowing” that I didn’t have enough information for it to be useful.

R^2 = r^2 + 25

So what? I surely needed to replace R with some kind of expression using r and a number… but I don’t have that information, so I have a ‘useless’ two variable equation and nothing else to work with.

I re-read the question and realised I’d been hung up with the seemingly useless application of Pythag instead of focusing on the fact that the question concerns area. Finally, I had that moment of clarity.

\pi R^2 = \pi r^2 + 25\pi

The bit I want is the shaded bit, so I can effectively ignore \pi r^2



Weird how it took me so long to ‘see’ it this time, and I spotted it so quickly a couple of years ago. I probably shouldn’t overthink that…