Two Triangle and Circle Problems
Several interesting geometry problems about triangles and circles came in recently. We’ll look at two today, and a third next week.
Trigonometry
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 …
Euler’s Formula: Complex Numbers as Exponents
Last week we explored how the polar form of complex numbers gives multiplication a simple geometric meaning. Here we’ll go one more step, and express polar form exponentially, which makes DeMoivre’s theorem trivial, and gives us a simple notation to replace “cis”.
Complex Numbers: Multiplication and Rotation
Having looked at the idea of complex numbers and how to perform basic operations on them, we are ready for one of the most important features for applications: their relationship to rotation. We’ll see this first in describing complex numbers by a length and an angle (polar form), then by discovering the meaning of multiplication …
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.
Subtleties of Inverse Trig Functions
(A new question of the week) It is not uncommon for students to ask about why they get different answers using different methods. Usually the answer is that the answers are really equivalent. This time, the answers really are different! This was partly the result of being taught an incomplete technique, omitting important cautions. And …
A Tunnel Through the Earth
I have a very short problem this week: How deep will you go if you dig a straight tunnel through the earth, how long will it be, and what angle do you have to start at?
Two Sinusoidal Models
(A new question of the week) Two recent questions involved using trigonometric functions to model real-life (or nearly so) situations, one about breathing, the other about a Ferris wheel. Both can be done by writing a sinusoidal function; the second can be done in other interesting ways as well.
Trig Identities: Where’d That Come From?
(A new question of the week) Proving a trigonometric identity can be a challenge; sometimes even when we read someone else’s proof, we can fail to see how they came up with a seemingly magical step. We’ll look at two such identities here, and consider how to bridge a gap when we are stuck.
Solving a Triangle: What Went Wrong?
Trigonometry can be a powerful tool for solving sides and angles in triangles. But you have to be careful with it! We’ll look at a classic type of error in solving an SSA triangle, get three explanations, and then see how knowing the context of a question can change our answer – to the point …