Ratios and Areas: An Unusual Pie Chart Problem
Here is a short problem with several levels of difficulty. The problem itself is poorly designed, as we’ll see, but still provides several useful lessons, dealing with measurement, rounding, and ratios.
When Newton’s Method Fails – and Why It’s So Good When It Doesn’t!
Last time, we saw how Newton’s method works. Here, we’ll look at a question about why it might not work, which will lead to a deep examination of how iterative methods work in general, from which we will discover why Newton’s method is as good as it is. I have to say, as I read …
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Solving Equations with Newton’s Method
Last time we solved some of the equations connected with a segment of a circle using Newton’s Method. Let’s take a closer look at the method – how it works, why it works, and a few caveats.
How to Find Any Part of a Segment of a Circle
Recent questions have dealt with calculations of various parts of a segment of a circle (chord, arc, sagitta, etc.). How, for example, can you find the arc length if you know the chord length and the height? The definitive explanation of these questions is found in a classic page from Ask Dr. Math, written by …
Proving a Radical Expression Is Rational
It can be tricky deciding how to approach a proof; this problem, whose answer requires going in a very different direction than you might expect, provides some interesting insights into the nature of proof. The proof itself, in fact, is far less interesting than the process of getting there!
Turning a Maximization Problem Inside-Out
Here is an interesting question we got recently, that turns a common maximization problem (the open-top box) inside-out. What do you do when you’re given the answer and have to find the problem? We’ll hit a couple snags along the way that provide useful lessons in problem-solving.
Inverse Trig Problems With Unstated Restrictions
Here are a pair of problems related to the last of the three we looked at last time, involving inverse trigonometric function identities with subtle issues. They were probably intended to be rather simple problems (though going beyond what I typically see in American classes) by ignoring the subtleties, but if you approach them as …
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Pitfalls of Inverse Trig Functions
A couple recent questions involved errors made both by students and by the authors of their textbooks, involving trigonometric or inverse trigonometric functions. These offer some good lessons in pitfalls to be aware of.
How to Evaluate Trig Functions (By Hand?)
In discussing the value of radians, we introduced the idea that trig functions are easier to evaluate that way. That raises the question, how do you find the value of a trigonometric function without a calculator, and how do calculators themselves do it? Let’s look into that.
Radians: Why, and When, They Are Better
A recent question reminded me that we hadn’t yet covered the topic of radians yet. We’ll look at several questions comparing radians to degrees, concluding with the recent question: Is a radian a unit, or something else?