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 …
(A new question of the week) Some kinds of problems can be solved at various levels; in particular, when we get a problem about ratios, we often can’t be sure whether the student knows only arithmetic, or can use algebra, which usually makes the problem easier. This is one reason we ask for a category …
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 …
Having just looked at the Law of Sines and the Law of Cosines, let’s consider how they can be applied to solving an oblique triangle – that is, finding missing parts of a triangle that is not a right triangle. The Ask Dr. Math site’s Trigonometry FAQ includes a concise summary of a procedure for …
Last week we looked at several proofs of the Law of Sines. Here we will see a couple proofs of the Law of Cosines; they are more or less equivalent, but take different perspectives – even one from before trigonometry and algebra were invented! Proof using coordinates First, here is a question we looked at …
(An archive question of the week) I’m in the middle of discussing the Law of Sines and the Law of Cosines, and in searching for questions about them, I ran across one that stands by itself. A student asks his teacher why his method without trig doesn’t work, and gets three answers from us. They …
Two of the most useful facts in trigonometry are the Law of Sines and the Law of Cosines (sometimes called the Sine Rule or Sine Formula, and the Cosine Rule or Cosine Formula). Over the years we were often asked where they come from (or are just asked about them, and reflexively offer proofs). There …
(A new question of the week) I enjoy getting questions from young children, as we did here. It forces us to try to express big ideas in simple words (or at least help their parents or teachers do so). A frequent subject of those questions is infinity – they seem fascinated by this concept, perhaps …
We’ve looked at the volume of a pyramid, the formula for which can be found geometrically by a couple very different methods. Cones can be handled the same way, so we can skim over them. Let’s finish up by considering the surface areas of both cones and pyramids. Volumes and areas I’ll start with a …