# The Flying Colours Maths Blog: Latest posts

## Ask Uncle Colin: A STEP vectors problem

Dear Uncle Colin, I’m struggling with this STEP question. The first two parts are fine — equality holds when there is some constant $k$ for which $a = kx$, $b = ky$ and $c=kz$, and part (i) follows directly from the original inequality. I can get an answer to part

## Why $\phi^n$ is nearly an integer

This article is one of those ‘half-finished thoughts’ put together late at night. Details are missing, and — in a spirit of collaboration — I’d be glad if you wanted to fill them in for me. The estimable @onthisdayinmath (Pat in real life) recently posted about nearly-integers, and remarked that

## Ask Uncle Colin: What is $\tan(1º)$?

Dear Uncle Colin, In my trigonometry homework, I’ve been asked to find the exact value of $\tan(1º)$. I have no idea where to start. Help! — A Nice, Gentle Little Exercise Hello, ANGLE. That’s an evil question, on many levels. It’s doable, but it’s not at all easy. In this

## The Maths Police Investigate: For The Love Of God, Make It Stop

The cadets are at it again. Thanks to @dragon_dodo and @FennekLyra. Agent Lyra and Agent Dodo were bored. After several weeks of suspending Gale’s desk in bizarre positions, fencing on the office chairs and sneakily pilfering Beveridge’s ginger beer, they had quite run out of things to do. Draped over

## Ask Uncle Colin: Can you prove $\sin(2x) \equiv 2\sin(x)\cos(x)$?

Dear Uncle Colin, I find it easier to remember trigonometric identities if I can ‘see’ how they fit together. I’m expected to know that $\sin(2x) \equiv 2\sin(x)\cos(x)$, but haven’t been able to prove it. Any ideas? — Geometry? Right Angles? How About Medians? Hi, GRAHAM! My favourite proof jumps out

## Brutal simultaneous equations

I recently became aware of the IYGB papers, available from Madas Maths. Like the Solomon papers, they’re intended to stretch you a bit — they’re ranked by difficulty from standard to extremely hard. My student, being my student, demanded we go through one of the extremely hard ones. There were

## Ask Uncle Colin: Why does the line with equation $10y+36x=16.5$ have a gradient of -3.6?

Dear Uncle Colin, I’ve got a line with equation $10y+36x=16.5$. That equation has no negative numbers in it, yet its gradient is apparently negative. I don’t understand why. — Silly Line, Only Positive Equation Dear SLOPE, It looks like we’re in misconception-land! In fact, you can write the equation of

## A surd simplification masterclass

The estimable @solvemymaths tweeted, some time back: hmm, perhaps I'll keep this one as "sin(22)" pic.twitter.com/cT5IHonoyb — solve my maths (@solvemymaths) January 16, 2016 A sensible option? Perhaps. But Wolfram Alpha is being a bit odd here: that’s something that can be simplified significantly. (One aside: I’m not convinced that

## Ask Uncle Colin: How did they get $\ln(50)$?

Dear Uncle Colin, I get $-\frac{\ln(0.02)}{0.03}$ as my answer to a question. They have $\frac{100\ln(50)}{3}$. Numerically, they seem to be the same, but they look completely different. What gives? — Polishing Off Weird Exponents, Really Stuck Dear POWERS, What you need here are the log laws (to show that $-\ln(0.02)=\ln(50)$,

## The Bigger Fraction

Some while back, Ben Orlin of the brilliant Maths With Bad Drawings blog posted a puzzle he’d set for some eleven-year-olds: Which is larger, $\frac{3997}{4001}$ or $\frac{4996}{5001}$? Hint: they differ by less than 0.000 000 05. He goes on to explain how he solved it (by considering the difference between