33
votes
Accepted
How important is knowledge of trig identities for use in Calculus
The specific identity
\begin{equation}\tag{A}
\tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x}
\end{equation}
as such is probably not often encountered, but simplifications akin to \...
- 5,879
22
votes
Why we don't normally teach chord, versine, coversine, haversine, exsecant, excosecant any more?
More of a comment than an answer but: They are all composites of more basic functions. In fact, all of the trig functions could can expressed in terms of sine, linear changes in coordinates, and ...
- 25.8k
20
votes
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Real-life problems involving solving triangles
There are many real-life problems involving solving triangles in "Trigonometry for the Practical Man" by James Edgar Thompson. Here are some from Chapter 5:
Problem 1
From the top of a ...
- 612
17
votes
Accepted
Memorizing Trig Identities
I agree this memorization is not necessary.
If students understand how the trigonometric functions are defined (unit circle) and know several basic identities, everything else can be derived. I think ...
- 286
16
votes
Accepted
Why is it written $\tan^{-1}$?
It's the functional inverse rather than the multiplicative inverse. Somewhere along the way, notation arrived at function composition being written as $f(f(x))=f^2(x)$, so the functional inverse ...
- 5,903
15
votes
Accepted
Is $180^\circ = \pi$?
It is possible to treat degrees and radians as units, just as we treat inches and centimeters as units. Just as we can convert inches to centimeters (1 in = 2.54 cm), we can convert degrees to ...
- 3,131
15
votes
How important is knowledge of trig identities for use in Calculus
Due to low enrollment, my AP Calc class was filled with the students who otherwise would have taken Pre-Calc this year. So you can imagine that "How much do you really need to know to see the bigger ...
- 5,637
13
votes
Why teach reference angles?
I disagree with your statement on the usefulness of reference angles, they have more use in the plane of the unit circle. So to answer the question specifically, yes we should still teach them and one ...
- 540
13
votes
Accepted
Why we don't normally teach chord, versine, coversine, haversine, exsecant, excosecant any more?
As several respondents have indicated, one could choose many different trigonometric functions to serve as the basic elements in terms of which other trigonometric functions are expressed and in (...
- 5,879
13
votes
Interesting Trigonometry problems
When introducing the sum of angle equations, I practiced presenting this. Starting with introducing a right triangle inside a rectangle. (I misplaced my notes, in which I had every step clearly laid ...
12
votes
Why do standard geometry textbooks not start with trigonometry?
If you really want to understand why the curriculum is structured the way it is, and how it got that way, you might want to read a brief history of the discourse around geometry education for the past ...
- 17.4k
11
votes
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Solving a Polar set of equations algebraically?
This might not quite be an "algebraic" solution in the sense the student was looking for, but I think a more reasonable approach to this sort of problem is using bounds, not setting things equal from ...
- 4,903
10
votes
Memorizing Trig Identities
If you want to integrate $\sin^m x$ or $\cos^m x$ for even $m$, you need to reduce the powers, which requires some trigonometric identities beyond pythagorean identities (in this case, $\sin^2 \theta =...
- 21.9k
10
votes
May we permit identities to be established by equivalent equations?
Well, if their "equivalences" are in fact equivalences,
then everything is fine.
The problem seems to be mostly a psychological one:
When students want to verify an equivalence, they check
the left-...
- 591
10
votes
Accepted
Is Plane Trigonometry by S. L. Loney still good as a textbook today?
I started my career as an archaeologist before I ended up in mathematics. I say this in order to emphasize that I am interested in trying to understand how people thought in the past—the usual "...
- 8,463
10
votes
Why is it written $\tan^{-1}$?
I will start by noting that I have written an answer to a similar question on Mathematics SE, and would direct you towards that answer for some additional discussion.
In my precalculus class, I tend ...
- 8,463
9
votes
Should word problems be reasonable?
Physically (or even economically) unreasonable answers are confusing. The student needs to be shown that math is a tool that can solve practical problems. Answers that are unreasonable send the ...
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9
votes
Symmetry in polar functions - how to explain
The problem underlying the discussion in the question can be summarized as that it is necessary to choose a branch cut to define a complex logarithm (or arctangent).
It is a mistake and pedagogically ...
- 5,879
9
votes
Is Trigonometry done differently in the US?
As you've tagged this as precalculus, I'll say that there is no standard precalculus curriculum in the United States and very little "official" guidance about what such a curriculum should ...
- 5,637
9
votes
What is this symbol called?
That's lowercase Greek phi, pronounced with an initial f sound and rhyming with English pie, lie or sky. That said, some speakers may pronounce it rhyming with English see, key or me.
See https://en.m....
- 5,042
9
votes
What is this symbol called?
Please check the entire Greek alphabet, as you can find it in this Wikipedia page: https://en.wikipedia.org/wiki/Greek_alphabet.
I must add that there are two ways to write the letter $phi$ in MathJax ...
- 2,237
8
votes
May we permit identities to be established by equivalent equations?
When you solve an algebraic equation like $2x - 4 = 0$, what you're really doing is finding assignments for the variables assuming the given equation is true. Doing the same thing to both sides of an ...
8
votes
Memorizing Trig Identities
If you have lots of time then you can surely derive most of the identities from the basics but basically it slows down your learning when these identites are required. Also not memorising these will ...
- 181
8
votes
Why do standard geometry textbooks not start with trigonometry?
I cannot speak to curricula outside of the US, and I don't really buy your argument that geometry courses are "focused more on memorizing theorems rather than understanding where they come from." It ...
- 8,463
8
votes
How to intuitively understand how the trig ratios are calculated
If you are teaching this at an introductory level, then the algorithm that calculators use today is going to go far over their heads. (It might go over MY head!) The story of how we developed ...
- 5,637
8
votes
How to layout a solution to a trig equation?
In most mathematics classes, we don't actually care about the solution to an exercise. The point is to get students to practice with the concepts, and figure out how to communicate their thinking. ...
- 8,463
8
votes
What's the Deal with Inverse Cotangent?
For teaching, I think this situation gives a good excuse to talk about how in mathematics, sometimes we get to embrace ambiguity.
One thing that differentiates the two definitions is that they satisfy ...
- 11.1k
8
votes
Real-life problems involving solving triangles
I’ve got one very specific answer to this question, because I had to dust off my old trigonometry knowledge just yesterday. I tell it as a real-world use of what I had to learn in school out of left ...
- 181
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