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?
Trigonometry
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 …
Trigonometry, Radicals, and a Very Common Error
(A new question of the week) While I was looking through recent questions to choose one to post, I ran across one that deals with an error we see very commonly – in fact, a student I had worked with that very afternoon in face-to-face tutoring had done the same sort of thing. The context …
A Triangle Problem: With Trig, and Without
(A new question of the week) When you are given a problem about a triangle, there can be many ways to approach it: pure geometry, trigonometry, and analytic geometry come to mind. When the context doesn’t dictate a method (as turns out to be true here), you just have to try what feels right to …
Application of Vectors: Airplane in the Wind
A recent question about the resultant velocity of an airplane illustrates different ways to make a diagram showing the bearings of air velocity and wind velocity, and to work out angles without getting too dizzy.
Sines Without Right Triangles
(A new question of the week) A recent question dealt with an apparent conflict between the right-triangle definition of sines and cosines, and the unit-circle definition, pertaining to multiples of 90° (angles on the axes). This provides an opportunity to look closely at the relationship between those two definitions. Two definitions Recall that the right-triangle …
Distances on Earth 3: Planar Approximation
We’ve looked at two formulas for the distance between points given their latitude and longitude; here we’ll examine one more formula, which is valid only for small distances. This is a “flat-earth approximation” to distance.
Distances on Earth 2: The Haversine Formula
Last week we started a series about finding distances on a sphere (which approximates the shape of the earth), using a straightforward formula from spherical geometry. But in practice, that formula turns out not to be ideal, so a different formula is used when accuracy in all circumstances matters. That is this week’s topic: first …