A Challenging Triangle Trigonometry Problem
(A new question of the week) Trigonometry identities can be hard to prove, and more so when they are specifically about a triangle.
Strategies
Comparing Logarithms With Different Bases
Logarithms are not hard to work with when only one base is involved (as in most real-life problems); but they can be challenging when each log has a different base. Here, we’ll look at a few problems in which we have to compare logarithms with different bases, showing various strategies.
Experimenting with Triangles and Circles
(A new question of the week) I intended to fit three problems into last week’s post, but the third was too interesting to shorten, so I’m posting it separately. The problem itself is not hard, but in looking for a more direct solution, we extend it, discovering (through geometry software) more general facts, which lead …
Two Triangle and Circle Problems
(A new question of the week) Several interesting geometry problems about triangles and circles came in recently. We’ll look at two today, and a third next week.
Three Trigonometric Inequalities
(A new question of the week) We often solve basic trigonometric equations; but a recent set of questions dealt with challenging trigonometric inequalities, which bring with them a new set of issues. We’ll look at several of those here, which combine trig with polynomials, rational functions, and more. Each will illustrate something new to watch …
Writing a Riemann Sum as an Integral
(A new question of the week) Riemann sums are used in defining the definite integral. But they can also be used in reverse: Sometimes you can be given the limit of a summation and asked to read it as a Riemann sum, and then turn it into an integral. Usually this is fairly straightforward; but …
A Rational Equation, With and Without Extraneous Roots
(A new question of the week) Extraneous roots can not only confuse the final solution to a problem; they can also make it harder to solve in the first place if you don’t deal with them early. Here is a relatively complicated rational equation, two questions about its solutions, and several ways to make it …
A Rational Equation, With and Without Extraneous Roots Read More »
Finding a Symmetric Polynomial Given Others
A recent question (whose asker refused to cooperate by showing work, so that we were unable to help) reminded me that we haven’t yet shown a similar kind of problem that can be quite interesting: problems where we are given the value of one or several expressions in several variables, and are asked for the …
Using the Angle Sum and Difference Identities
Here is a short discussion of a common type of problem in trigonometry classes: finding a trig function of the sum or difference of two angles, given minimal information about them.
Advanced Logarithm Problems
(A new question of the week) Here is an interesting collection of problems involving logarithms with different bases, which require some unique thinking. And after we’d worked out a good strategy, another problem arose at a whole new level.