For the next couple weeks, we’ll look at a few “dissection problems”, which involve cutting a shape in various ways. Here, we look at what turns out to be a simple problem, though it seems at first that it would be complicated: dividing a cake into pieces with the same amount of cake and the …
(A new question of the week) Usually when we have a figure labeled with some lengths and angles, we can expect to find unknown angles using trigonometry. When we are expected to do this using geometry alone, we can expect that there is something special about the figure that makes it possible. But how to …
(A new question of the week) A question in September, about graphing a horizontally-stretched cosine function, led to a long conversation. Between a typo in the problem and some inside-out thinking, this surprisingly non-routine problem led to some good mind-stretching! I have edited this down considerably by removing distractions from the main ideas, but it …
A Mind-Stretching Exercise with a Stretched Cosine Read More »
(A new question of the week) We have had a lot of interesting questions recently. This one involved inverse trigonometric equations that led to cubic and quartic equations. We’ll observe here one of the benefits of embedding the original discussion in a blog format where I can add information that will help you, the reader, …
Challenging Inverse Trig and Polynomial Equations Read More »
Having covered the basics of defining and adding vectors, multiplying by scalars and finding unit vectors, it’s time to look at multiplying vectors together. What makes this entirely unlike working with numbers is that there are two ways (in fact, more than two!) to multiply two vectors. We’ll look at one of those today, the …
We’re looking at the concept of vectors at an introductory level. Last week we looked at how they are defined in this context (as quantities with magnitude and direction), and how they are added (which is really part of the definition). Our collection of answers from Ask Dr. Math this time focuses on the ideas …
We have been looking at questions about the roundness of the earth, starting with the general fact, and then the determination of the size of the earth. A very common question is about how that roundness affects what we can see, sometimes as a challenge (“If I can see this, then how can the earth …
(A new question of the week) A recent series of questions from one student involved interesting combinations of trigonometric identities and solutions of polynomials. At one time using trig to solve equations was far better known than it is today, and these presumably are meant in part as an introduction to those ideas. Some were …
Having just looked at how to solve oblique triangles, let’s look at a couple “word problems” (applications) involving such triangles. We’ll be using the Law of Sines, and also exploring alternative methods of solution. A tilted tree Let’s start with this real application from 1999: Will the Tree Hit the House? There is a tree …
Last time we looked at solving triangles in the ASA, AAS, SSS, and SAS cases. We have one more case, which tends to be a little more complicated: the “ambiguous case”, SSA. SSA: Two sides and the angle opposite one of them We’ll start with this question from 1998: Triangles and Law of Sines I …