# Alternatives

## Function Transformations Revisited (I)

(A new question of the week) Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways. A recent discussion brought out some approaches that nicely supplement what we have said before. Here, the focus will be on …

## Solving a Triangle: What Went Wrong?

Trigonometry can be a powerful tool for solving sides and angles in triangles. But you have to be careful with it! We’ll look at a classic type of error in solving an SSA triangle, get three explanations, and then see how knowing the context of a question can change our answer – to the point …

## A Triangle Problem: With Trig, and Without

(A new question of the week) When you are given a problem about a triangle, there can be many ways to approach it: pure geometry, trigonometry, and analytic geometry come to mind. When the context doesn’t dictate a method (as turns out to be true here), you just have to try what feels right to …

## Summing Squares: Finding or Proving a Formula

Last week we looked at problems about counting the squares of all sizes in a checkerboard. Some solutions required finding the sum of consecutive squares, $$1^2+2^2+3^2+\dots+n^2$$, for which we used a formula whose derivation I deferred to this week. Here we’ll see a couple proofs that require knowing the formula ahead of time, and a …

## Application of Vectors: Airplane in the Wind

A recent question about the resultant velocity of an airplane illustrates different ways to make a diagram showing the bearings of air velocity and wind velocity, and to work out angles without getting too dizzy.

## How Many Paths from A to B?

A popular kind of question in combinatorics is to count the number of paths between two points in a grid (following simple constraints). This can be done by very different methods at different levels. We’ll look at several problems of this type, starting with the simplest.

## Sines Without Right Triangles

(A new question of the week) A recent question dealt with an apparent conflict between the right-triangle definition of sines and cosines, and the unit-circle definition, pertaining to multiples of 90° (angles on the axes). This provides an opportunity to look closely at the relationship between those two definitions. Two definitions Recall that the right-triangle …

## Multiplying Fractions by Whole or Mixed Numbers

Last week we looked at how to multiply fractions, and why we do it that way. But what do we do when one of the numbers is a whole number, or when one or both are mixed numbers? And do we have to do it the way we are taught?

## Angles in a Star

(A new question of the week) I like problems that can be solved in multiple ways, which can train us in seeing the world from different perspectives. Late in November we dealt with a pair of such questions involving angles in star-like figures.

## A Geometrical Limit

(A new question of the week) We usually see limits applied to functions in a calculus class. An interesting question from late October deals with a limit in a geometrical construction based on a function. We’ll be seeing how to discover a proof, then several alternative proofs, and finally what the answer means.