# The Flying Colours Maths Blog: Latest posts

## Ask Uncle Colin: Some missing solutions

Dear Uncle Colin, When I solve $2\tan(2x)-2\cot(x)=0$ (for $0 \le x \le 2\pi$) by keeping everything in terms of $\tan$, I get four solutions; if I use sines and cosines, I get six (which Desmos agrees with). What am I missing? – Trigonometric Answers Not Generated – Expecting 'Nother Two

## Wrong, But Useful: Episode 46

In this month's edition of Wrong, But Useful, @reflectivemaths and I are joined by special guest co-host @dragon_dodo, who is Dominika Vasilkova in real life. We discuss: What maths appeals to a physicist. Dominika's number of the podcast: $0.110001000000000000000001…$, Liouville's constant, which is $\sum_{n=1}^\infty 10^{-n!}$, the first constant to be

## A coin sequence conundrum

Zeke and Monty play a game. They repeatedly toss a coin until either the sequence tail-tail-head (TTH) or the sequence tail-head-head (THH) appears. If TTH shows up first, Zeke wins; if THH shows up first, Monty wins. What is the probability that Zeke wins? My first reaction to this question

## Ask Uncle Colin: A Polar Expression

Dear Uncle Colin, I was asked to find the tangent to the curve $r=\frac{8}{\theta}$ at the point where $\theta = \frac{\pi}{2}$. I worked out $\dydx = \frac{ \frac{8 \left(\theta \cos(\theta)-\sin(\theta)\right)}{\theta^2}}{\frac{-8\left(\theta \sin(\theta)+\cos(\theta)\right)}{\theta^2} }$, which simplifies to $-\frac{\theta \cos(\theta)-\sin(\theta)} {\theta \sin(\theta)-\cos(\theta)}$. Evaluated at $\theta = \frac{\pi}{2}$, that gives $\dydx=\frac{2}{\pi}$ and a

## The Mathematical Ninja and Cosines

As the student was wont to do, he idly muttered "So, that's $\cos(10º)$…" The calculator, as calculators are wont to do when the Mathematical Ninja is around, suddenly went up in smoke. "0.985," with a heavy implication of 'you don't need a calculator for that'. As the student was wont

## Ask Uncle Colin: Factors!

Dear Uncle Colin, If you know all of the factors of $n$, can you use that to find all of the factors of $n^2$? For example, I know that 6 has factors 1, 2, 3 and 6. Its square, 36, has the same factors, as well as 4, 9, 12,

## A common problem: not reading carefully

I'm a big advocate of error logs: notebooks in which students analyse their mistakes. I recommend a three-column approach: in the first, write the question, in the second, what went wrong, and in the last, how to do it correctly. Oddly, that's the format for this post, too. The question

## Ask Uncle Colin: Trouble in Sector ABC

Dear Uncle Colin, I got stuck on this sector question, which asks for the radius of circle $P$, which touches sector $ABC$ as shown. I’m given that $ABC$ is a sector of a circle with centre $A$ with radius 12cm, and that angle $BAC$ is $\frac{\pi}{3}$. My answer was 3.8cm,

## Attack of the Mathematical Zombies: Calc vs non-calc

Another horde of zombies lumbered into view. "What are they saying?" asked the first, readying the shotgun as he'd done a hundred times before. "Something about the calculator exam," said the second. "It's hard to make out." He pulled some spare shells from his bag. "Calculator papers are easier!" groaned

## Ask Uncle Colin: A STEP in the right direction

Dear Uncle Colin, I'm struggling with a STEP question. Any ideas? Given: 1. $q^2 – pr = -3k$ 2. $r^2 – qp = -k$ 3. $p^2 – rq = k$ Find p, q and r in terms of k. – Simultaneous Triple Equation Problem Hi, STEP, and thanks for your