# Browsing category ask uncle colin

## Ask Uncle Colin: A dangling rope

A Hanging Rope Dear Uncle Colin, I'm designing a small cathedral and have an 80-metre long rope I want to hang between two vertical poles. The poles are both 50 metres high, and I want the lowest point on the rope to be 20 metres above the ground. How far

## Ask Uncle Colin: Integrating $\sec$ and $\cosec$

Dear Uncle Colin, I keep forgetting how to integrate $\sec(x)$ and $\cosec(x)$. Do you have any tips? - Literally Nothing Memorable Or Distinctive Hi, LNMOD, and thanks for your message! Integrating $\sec(x)$ and $\cosec(x)$ relies on a trick, and one the average mathematician probably wouldn't come up with without a

## Ask Uncle Colin: A Strange Simultaneous Equation

Dear Uncle Colin, I have the simultaneous equations $3x^2 - 3y = 0$ and $3y^2 - 3x = 0$. I've worked out that $x^2 = y$ and $y^2 = x$, but then I'm stuck! - My Expertise Relatedto1 Simultaneous Equations? Not Nearly Enough! Hi, MERSENNE, and thanks for your message!

## Ask Uncle Colin: A Factorial Sum

Dear Uncle Colin, I have been given the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{8} + \frac{1}{30} + \frac{1}{144} + ...$, which appears to have a general term of $\frac{1}{k! + (k+1)!}$ - but I can't see how to sum that! Any ideas? - Series Underpin Maths! Hi, SUM, and thanks

## Ask Uncle Colin: Some Ugly Trigonometry

Dear Uncle Colin, How do I verify the identity $\frac{\cos(\theta)}{1 - \sin(\theta)} \equiv \tan(\theta) + \sec(\theta)$ for $\cos(\theta) \ne 0$? - Struggles Expressing Cosines As Nice Tangents Hi, SECANT, and thanks for your message! The key questions for just about any trigonometry proof are "what's ugly?" and "how can I

## Ask Uncle Colin: A factorising trick

Dear Uncle Colin, How would you factorise $63x^2 + 32x - 63$? I tried the method where you multiply $a$ and $c$ (it gives you -3969) - but I'm not sure how to find factors of that that sum to 32! Factors Are Troublesomely Oversized, Urgh Hi, FATOU, and thanks

## Ask Uncle Colin: A Mess of Logs

Dear Uncle Colin, I have to show that $-\frac{x}{2} = \ln (\sqrt{1+e^x} - \sqrt{e^x }) + \ln (\sqrt{1+e^{-x}} + 1)$. I can't get it anywhere near the right form! - Proof Of It Not Coming - Any Reasonable Explanation? Hi, POINCARE, and thanks for your message! That's a bit of

## Ask Uncle Colin: A Cubic That Won’t Come Good

Dear Uncle Colin, I'm told that $x\sqrt{x} - 5\sqrt{x} = 2$ and I have to find $x - 2\sqrt{x}$. Everything I try seems to make it worse! Any ideas? Mastering A Cubic - Help Is Needed Hi, MACHIN, and thanks for your message! At first glance, that's a strange one.

Dear Uncle Colin, As I progress through my maths education, I notice that the people around me are getting smarter and smarter. How do I keep my head up when everyone is brighter than me? I'm Mightily Put Off Seeing Their Outstanding Results Hi, IMPOSTOR, and thanks for your message!

## Ask Uncle Colin: Why is it called “completing the square”?

Dear Uncle Colin, Why is it called "completing the square"? To me, it looks like you're taking something away from a square. - Some Quadratics, Understandably, Are Requiring Explanation Hi, SQUARE, and thanks for your message! Completing the square involves taking a quadratic such as $x^2 + 6x + 5$