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33 votes
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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 \...
Dan Fox's user avatar
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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 ...
Steven Gubkin's user avatar
20 votes

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 ...
Mike Z.'s user avatar
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17 votes
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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 ...
martinkunev's user avatar
16 votes
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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 ...
Adam's user avatar
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15 votes
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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 ...
Andreas Blass's user avatar
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 ...
Matthew Daly's user avatar
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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 ...
Jeffery Thompson's user avatar
13 votes
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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 (...
Dan Fox's user avatar
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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 ...
JTP - Apologise to Monica's user avatar
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 ...
mweiss's user avatar
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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 ...
Daniel Hast's user avatar
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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 =...
Chris Cunningham's user avatar
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-...
Uwe's user avatar
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10 votes
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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 "...
Xander Henderson's user avatar
  • 8,298
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 ...
Xander Henderson's user avatar
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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 ...
guest's user avatar
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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 ...
Dan Fox's user avatar
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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 ...
Matthew Daly's user avatar
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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....
J W's user avatar
  • 4,893
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 ...
Dominique's user avatar
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8 votes
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Rules to eliminate erroneous solutions in Trig equations?

This isn't specific to trigonometry at all; it's really about understanding what it means to "solve an equation". Given functions $f, g: \mathbb{R} \to \mathbb{R}$, to solve the equation $f(x) = g(x)$ ...
Daniel Hast's user avatar
  • 4,893
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 ...
Kartik's user avatar
  • 181
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 ...
YawarRaza7349's user avatar
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 ...
Xander Henderson's user avatar
  • 8,298
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 ...
Matthew Daly's user avatar
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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. ...
Xander Henderson's user avatar
  • 8,298
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 ...
user52817's user avatar
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