A Challenging Triangle Trigonometry Problem
(A new question of the week) Trigonometry identities can be hard to prove, and more so when they are specifically about a triangle.
… And The Oldest Has Red Hair
It’s been a while since we’ve done a puzzle, just for fun. Here we’ll look at two versions of a riddle, about finding children’s ages from a known product, a partially known sum, and a bizarre fact about the oldest. Then we’ll close with an interesting variation.
Monotonic Functions, Inequalities, and Optimization
Looking for a cluster of questions on similar topics, I found several from this year in which monotonic functions (functions that either always increase, or always decrease) provide shortcuts for various types of problems (optimization with or without calculus, and also algebraic inequalities). We’ll look at a few of these.
Trig Terminology: What Do Those Words Mean?
Students sometimes wonder why the trigonometric functions (sine, cosine, tangent, secant, and so on) have the names they do, and how they relate to the corresponding terms in geometry. How are the tangent and secant functions related to tangent and secant lines in trigonometry? And what in the world is a sine? Here we’ll look …
Inverse Trig Notation: What Do sin^-1 and arcsin Mean?
Since we’ve been looking at an example of ambiguity in notation, let’s look at a very different one. There is a lot to be confused by in inverse trigonometry! We’ll try to untangle the notations of \(\sin^{-1}\) and \(\arcsin\).
Implied Multiplication 3: You Can’t Prove It
This is the last of a series on our discussions, since I closed comments at the end of 2021, of Implied Multiplication First (IMF), the idea that multiplications written by juxtaposition, rather than with a symbol, are to be done before other multiplications or divisions. Last time, we saw that there is no “official” answer. …
Implied Multiplication 2: Is There a Standard?
This is part 2 of a series of extracts from discussions we have had on whether multiplication implied by juxtaposition is to be done before division (which I call IMF, for Implied Multiplication First). Some people write to us claiming that there is one official correct answer. Are they right?
Implied Multiplication 1: Not as Bad as You Think
We keep getting new questions related to Order of Operations: Implicit Multiplication?, where we discussed expressions like 6/2(1+2) that keep showing up in social media arguments. Since I closed comments on that page some time ago, because of the toxicity of some of them, further questions have come through our Ask a Question page (as …
Implied Multiplication 1: Not as Bad as You Think Read More »
Zero Factorial: Why Does 0! = 1 ?
We’ve been talking about the oddities of zero, and I want to close with another issue similar to last week’s \(0^0\). All our questions will be essentially identical apart from details of context: “We know zero factorial equals 1; but why?” This isn’t nearly as controversial as the others, but will bring closure to the …
Zero to the Zero Power: Indeterminate, or Defined?
Last week we looked at numbers raised to the zero power, as part of our series on oddities of zero. We’ve looked at zero divided by zero in the past, and just recently observed how 0 to the 0 power relates to degree in polynomials, which is part of the motivation for this series. But …
Zero to the Zero Power: Indeterminate, or Defined? Read More »