Trigonometry

Order of Operations: Trigonometric Functions

(An archive question of the week) Last time we looked at some details that are rarely mentioned in stating the conventions for interpreting algebraic expressions. I couldn’t fit a discussion of the most complicated case: trigonometric functions, which when written without parentheses, as they traditionally have been, can raise several issues. (Much of the same …

The Art of Proving Trig Identities

(A new question of the week) Last week we looked at a recent question about basic trigonometric equations. That discussion continued into the subject of identities, which we’ll look at here. We’ll be sitting in on an extended chat about many important aspects of this kind of work. It’s still very long, even after extensive …

An Introduction to Trigonometry

(An archive question of the week) While I’m showing some recent explanations of basic trigonometry techniques, this is a good time to look at an even more basic explanation of the essentials of the subject for a beginner. Right triangle trigonometry Here is the question, from 2001: Trigonometry in a Nutshell I’m in 8th grade …

Trigonometric Equations: An Overview

(A new question of the week) This week and next I will look at a recent discussion on trigonometry that dug deep into two different issues: solving equations, and proving identities. These are good summaries of how to approach these common kinds of problems. This week: solving basic trig equations. Questions about trig functions Here …

Circumcircles and the Law of Sines

(A new question of the week) Today I want to look at a recent question that led into both geometrical and trigonometrical solutions, and particularly a useful perspective on the Law of Sines. The problem Here is the question, from March: A quadrilateral ABCD is put inside a circle with radius 1. AB = √3, …

A Triangle in a Semicircle

(A new question of the week) Like many questions we get, this one can be solved in many ways. We like to guide a student to whatever solution will fit what they have learned; along the way, we may find various additional methods, and side trips into other topics of interest. A two-part problem This …

Rewriting a Sum of Trig Functions

(A new question of the week) It has been a while since I have regularly included recent questions to this site, in part due to my focus on big topics. This summer I will be focusing on individual questions, some large and some small. Today, we’ll have two recent questions that look at the same …

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 …