# Browsing category ask uncle colin

## 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 $x^2 + 6x + 5$

Dear Uncle Colin, I struggled with a problem where you had 5 blue balls and x green balls, and the probability of picking two blue balls out of the bag was $\frac{5}{14}$. I can't really see where to start! -Baffled About Likelihood, Lacking Startpoint Hi, BALLS, thanks for your message!

## Ask Uncle Colin: A Set Square Mark

Dear Uncle Colin I just bought a new set square and noticed it had a couple of extra marks - one at seven degrees and one at 42 degrees. Have you any idea what those are for? - Don't Recognise Extra Information Engraved on Calculus Kit Hi, DREIECK, and thank

## Ask Uncle Colin: Some number theory

Dear Uncle Colin, I need to show that $\sqrt{7}$ is in $\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{7}]$ and I don't really know where to start. We Haven't Approached Tackling Such Questions Hi, WHATSQ, and thanks for your message! I am absolutely not a number theorist, although I must admit to getting a bit curious about

Dear Uncle Colin, I'm ok with my basic power laws, but I don't understand why $x^0$ is always 1, and I get mixed up when it's a fraction or a negative power. Can you help? Running Out Of Time Hi, ROOT, and thanks for your message! If it's any consolation,