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 \...
- 5,679
30
votes
Accepted
How to convince my student that this is an Identity : $\sec^2x-\tan^2x=1$?
An identity always holds for some values of the free variables. Sometimes the allowed values are all real numbers, sometimes something else. In this case the identity holds whenever $\sec x$ and $\tan ...
- 3,680
26
votes
Why do we conventionally treat trig functions as going anti-clockwise from the right?
Regarding the second part of your question:
Would I be doing my students a terrible disservice if I introduced them to trig functions treating them as going clockwise from 12 o'clock?
I think it's ...
- 17.1k
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 ...
- 23k
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
15
votes
Rigorously defining the concept of an angle for high school students
Many high school geometry textbooks define an angle as simply
the union of two rays with a common endpoint
The advantage of this definition is its simplicity. Among its disadvantages:
It does ...
- 17.1k
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 ...
- 5,569
14
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 ...
- 3,011
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 ...
- 530
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 (...
- 5,679
12
votes
Why do we conventionally treat trig functions as going anti-clockwise from the right?
My guess is that it comes from drawing complex numbers. Putting real numbers on the $x$-axis from left to right and imaginary numbers on the $y$-axis from bottom to top matches with the way we tend to ...
- 5,664
11
votes
Accepted
Why do we conventionally treat trig functions as going anti-clockwise from the right?
There is no such thing as a natural sign/direction convention until long after the fact. Consider another answerer's comment that anyone using the left hand rule to construct the cross product of two ...
- 641
11
votes
Accepted
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 ...
- 4,853
11
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 ...
- 17.1k
11
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 ...
10
votes
What purpose do these kinds of question serve in mathematical training?
I suspect the intended use of these questions was to provide practise using trigonometric identities and also the idea of arithmetic progression (which I think would both appear in the UK A-level ...
- 5,664
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 =...
- 20.4k
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-...
- 580
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 "...
- 7,137
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 ...
- 164
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 ...
- 5,679
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 ...
- 5,569
8
votes
Accepted
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 $f(x) = g(x)$ ...
- 4,853
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
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
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 ...
- 7,137
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. ...
- 7,137
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