# Trigonometry

## Area of a Plot of Land

The last four posts dealt with formulas for finding areas using lengths of sides, starting with the triangle, where that is all you need, and then quadrilaterals, where something more must be added; and then using coordinates of vertices. Now we can use those tools to solve some of the more common real-life problems we …

## Graphing Transformed Sines

I’ll close out our look at transformations of functions with some trigonometric graphs. These are the best example of combined transformations, and involve some special tricks as well. We’ll start with an early question that gives an overview of the process, then focus in on important details. Overview First, a typical question from 1997, to …

## Distances to an Arc: Exact and Approximate Formulas

(A new question of the week) It can be an interesting challenge to be presented with a formula and asked how it was derived. This becomes a bigger challenge when the formula is only approximate, so we have to figure out how to arrive at this particular approximation. But it is impressive when several different …

## Mixing Trig and Inverse Trig Functions

(A new question of the week) Since we’ve been doing a little trigonometry this week, let’s look at a recent set of questions from a student in the Philippines, all about compositions of trig and inverse trig functions. This is a topic with some interesting little surprises, and several of us answered, to give a …

## Measuring Water in a Tank

Over the years, we have received a huge number of questions asking about how to find the amount of liquid (water, oil, …) in a tank, usually a horizontal cylindrical tank. The simplest case involves a rather complicated formula; from there, we can reverse the formula (finding the depth for a given volume), or we …

(A new question of the week) In many areas of math, an answer can come in several forms, which can make it hard to know if you are right when you compare your answer to the answer in the back of the book. Even worse is when the problem is multiple-choice, and your answer has …

## Finding the Radius of a Sphere

(An archive question of the week) An interesting question came to us in 2016, where rather than using a well-known formula, it was necessary to work out both what data to use, and how to calculate the desired radius.

## What is Cosine – With a Twist

(A new question of the week) Having written last week about the definitions of trigonometric functions, I want to look at a question from a few months ago that illustrates a rather common mistake students make in applying those definitions. It also demonstrates the patience required to find out what is in a student’s mind, …

## Proving an Identity in Different Ways

(A new question of the week) Having discussed trigonometric identities on Monday, let’s make this Trig Week, by looking at a discussion from two months ago in which we were asked about alternative routes to a proof.

## 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 …