# Trig Identities #4 Double Angles

Starting with one of our ‘sum and difference’ identities:

$\sin(x\pm y)=\sin(x)\cos(y) \pm \cos(x)\sin(y)$

If we take $\sin(2x) = \sin(x+x)$

then, using the identity we started with:

$\sin(2x)=\sin(x)\cos(x) + \cos(x)\sin(x)$

$\sin(2x)= 2\sin(x)\cos(x)$

We can do it all again with cos:

$\cos(2x) = \cos(x+x)$

$\cos(2x) = \cos(x)\cos(x) - \sin(x)\sin(x)$

$\cos(2x)=\cos^2(x)-\sin^2(x)$

Now, recall that $\cos^2(x) +\sin^2(x) = 1$

$\cos^2(x) = 1 - \sin^2(x)$

so

$\cos(2x) = 1 - \sin^2(x)-\sin^2(x)$

$\cos(2x) = 1 - 2\sin^2(x)$

That’s one identity, now if we go back to the start and manipulate it all a bit differently:

$\cos(2x)=\cos^2(x)-\sin^2(x)$

$\cos(2x)=\cos^2(x)-(1-\cos^2(x))$

$\cos(2x)=\cos^2(x) - 1 +\cos^2(x)$

$\cos(2x)=2\cos^2(x) - 1$

Finally, tan is pretty straight forward if you use the sum&difference identity, but we haven’t derived that yet, so here we go:

$\tan(x)=\frac{\sin(x)}{\cos(x)}$

$\tan(x + y) = \frac{\sin(x+y)}{\cos(x+y)}$

Using the sin & cos sum/difference identities:

$\tan(x + y) = \frac{\sin(x)\cos(y) + \cos(x)\sin(y)}{\cos(x)\cos(y)-\sin(x)\sin(y)}$

$= \frac{\sin(x)\cos(y) + \cos(x)\sin(y)}{\cos(x)\cos(y)-\sin(x)\sin(y)} * \frac{\frac{1}{\cos(x)\cos(y)}} {\frac{1}{\cos(x)\cos(y)}}$

$= \frac{\frac{\sin(x)\cos(y)}{\cos(x)\cos(y)} + \frac{\cos(x)\sin(y)}{\cos(x)\cos(y)} } {\frac{\cos(x)\cos(y)}{\cos(x)\cos(y)} - \frac{\sin(x)\sin(y)}{\cos(x)\cos(y)}}$

$\tan(x + y) = \frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}$

Phew! Now, if x and y are equal, then just call them both x:

$\tan(2x) = \frac{\tan(x)+\tan(x)}{1-\tan(x)\tan(x)}$

$\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}$

# 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).

Hence:

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

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

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

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

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

$x^2+y^2=1$

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 $sin^2{\theta }$:

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

$cot^2(\theta)+1=csc^2(\theta)$

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

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

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

# Identity

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).”

625

• 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. 🙂