# Month: May 2018

## What Are Trig Functions, Really?

(An archive question of the week) Trigonometric functions are sometimes introduced without a deep explanation of their meaning; they are just buttons to push on a calculator, or names to write in an equation. Even when a textbook gives a careful presentation, there are so many facets to the concept that it can be easy …

## Different Ways to Prove a Trigonometric Identity

Proving trigonometric identities can be a major challenge for students, as it is often very different from anything they have previously done. Often they confuse this concept with solving an equation. But also, they may be give overly rigorous standards to comply with. Here, I will look at several discussions we have had about different …

## Two Solutions to an ODE

(New Question of the Week) This recent question involves an ordinary differential equation (ODE) and the relation between different solutions. It illustrates common difficulties in interpreting what a problem is asking for, as well as some communication problems involving language and notation.

## Integration by Substitution

(An archive question of the week) Last time, we looked at a method of integration, namely partial fractions, so it seems appropriate to find something about another method of integration (this one more specifically part of calculus rather than algebra). We will look at a question about integration by substitution; as a bonus, I will …

## Partial Fractions: How and Why

I have often noted that calculus class is where you really learn algebra. Certain techniques in calculus demand algebraic skills that either were not taught in algebra classes (because they are not needed until you get to calculus), or have been forgotten. Chief among these is the method of partial fractions. I have here put …

## What Do Exponents Mean?

(New Question of the Week) We recently had a long discussion about a very common question from a somewhat different perspective: What do exponents (zero, negative, fractional, …) actually mean? The hard part, in the end, was to decide what “mean” means. What does it mean to define something in math? I will pick out the main thread of the …

## Subtleties in a Logic Puzzle

(Archive Question of the Week) Logic puzzles can exercise our ability to reason carefully. Interestingly, the use of formal logic in doing so can actually get in our way, because such puzzles often have subtleties in their wording that are hard to capture in formal logic. Examining our thinking carefully can help us see wrong …

## Necessary and Sufficient Conditions: If, or Only If?

Sometimes in math, we trip over words, especially when they are used in ways that differ from everyday usage, or when the associated grammar is complicated. This set of three answers from our archive, each of which is referred to by the next one, look at relationships among the ideas of “necessary and sufficient conditions”, …

## Challenging Rate Questions

(New question of the week) A conversation last week went through a number of interesting questions, starting with a couple on percentages, and moving into some that I would call rate questions. I will extract these, which I think will be useful for others. (The rest could, too, but there was just too much there …

## Mathematical Thinking Solves an Operation Puzzle (Or Not)

(Archive problem of the week) Having just written about sequence puzzles, which sometimes can be solved mathematically, and sometimes are just psychological tests, I want to show a different kind of puzzle that I ran across while searching for those. At first, it looks like mere guess-and-check; then we find it can be solved easily …