Questions tagged [trigonometry]
For questions about effectively motivating and teaching the concepts of trigonometry, including the unit circle, the sin/cos/tan functions, and other related ideas.
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Pythagoras and Trigonometry sequencing
In teaching the high school curriculum Pythagoras is usually bundled with Trigonometry. They might be justified by way of proof of some sort. They are used to solve 2d and 3d geometry problems for ...
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Multiple proofs for the same problem
One way of encouraging students to explore mathematics can be letting them to use different approaches to solve the same problem. If students can find alternatives from different areas of mathematics ...
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Simple way to explain Sine theorem applications
what is the simplest way to explain how to determine whether the resolution of a triangle (finding all its sides and angles, given 3 of them) using the sine theorem gives one or two solutions and ...
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What's the Deal with Inverse Cotangent?
So, I was minding my own business and I thought I had defined inverse cotangent in the natural fashion. In particular, we define inverse tangent as the inverse of tangent restricted to $(-\pi/2, \pi/2)...
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How to teach relationship between slopes and angles?
I am trying to write a very very basic trig primer, from scratch.
Early on, I wish to discuss slopes (say, the slope of a line through the origin), and I would like to give the 'right' hints on how ...
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Is Trigonometry done differently in the US?
I'm Italian and I've watched some videos from Americans and noticed a weird thing. Let's talk about a linear trigonometric equation like this:
$$\sin x+\cos x+\sqrt3=0.$$
I've seen Americans solving ...
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Phase shift vs. horizontal shift, and frequency vs. angular frequency in sinusoidal functions
TL;DR version:
It seems to me that high school curricula no longer distinguish between "horizontal shift" and "phase shift", or between "frequency" and "angular ...
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Interesting Trigonometry problems
After explaining some basic trigonometry to my kid, such as $\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$, Law of sines, Law of cosines, I wonder if there are some ...
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How to layout a solution to a trig equation?
I am interested in how you would encourage students to layout their working to a trigonometric equation.
For instance, let's consider this problem:
Solve the equation $6\cos x - 8\sin x = 7$ for $0 &...
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Python programming - math library that uses degrees by default [closed]
Other than the standard math module for Python3, is there another library out there that uses degrees by default (as opposed to radians)? I am teaching students to use turtle (which uses degrees by ...
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How to intuitively understand how the trig ratios are calculated
I've asked a question on Math Stack Exchange, but it was suggested it might be a better idea to post it on this Educators instead.
Here's the question link:
https://math.stackexchange.com/questions/...
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Which textbooks on College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE are written by world-class mathematicians?
For example, Trigonometry was written by Wolf-Prize winner Israel Gelfand, one of the top mathematicians in the 20th century. I am wondering if other world-class mathematicians have written textbooks ...
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Symmetry in polar functions - how to explain
In the precalculus curriculum I am teaching (using Stewart's book Precalculus: Mathematics for Calculus, 7th ed.), we do a bit of polar graphing, which includes discussion of symmetry on polar graphs. ...
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How important is knowledge of trig identities for use in Calculus
I have a question regarding tutoring a calculus student. They need to prove trig identities such as $$\frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2\sec^2x.$$ Doing this kind of problem is very tedious and ...
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Is Plane Trigonometry by S. L. Loney still good as a textbook today?
I am considering using S. L. Loney's Plane Trigonometry as the textbook for my course in trigonometry and would like to ask for opinions about the book. This books is very odd and there might be pros ...
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Pedagogical considerations behind current order of presentation of trigonometry
A pre-calculus book (Precalculus ed 1 By Miller and Gerken), presents trigonometry in the following order:
1- Angles
2- Trigonometric functions defined on the unit circle
3- Right triangle ...
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Best Way to Learn Trigonometry
What are the best resources to learn trigometry? I recently decided to pursue a BS in mathematics at uni. I used to fail all my math classes with D's or F's until I started teaching myself, and so far ...
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How to formulate this type of arcsin problem?
Reading and commenting on What are some common ways students get confused about finding an inverse of a function? I was kindly set straight that the use of $\sin^{^{-1}}(x)$ to mean $\arcsin(x)$ has ...
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"Amplitude" of Tan and Cot functions
The amplitude of a sinusoid is the distance from its axis to a high
point or a low point.
When we read this, it follows that Tan and Cot don't have an amplitude. Nor do SEC or CSC. Now, I'm in an ...
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Why do standard geometry textbooks not start with trigonometry?
Throughout my geometry course, I was given many theorems and postulates, which I was were expected to memorize and apply. At the time, I sorta went along with it, but I couldn’t help but wonder where ...
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Is $180^\circ = \pi$?
I want to ask a question that causes confusion. In the trigonometry, we use some units of measure of angle: degree and radian. Which is/are correct?
$$ 180^{\circ} = \pi $$
or
$$ 180^{\circ} = \pi \...
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Why we don't normally teach chord, versine, coversine, haversine, exsecant, excosecant any more?
It seems that the following functions are not only excluded from a course in trigonometry, they are almost never taught in any course:
Chord
Versine
Coversine
Haversine
Exsecant
Excosecant
I could ...
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Why do we state the antiderivative of $\sec x$ as $\ln |\sec x +\tan x|+C$?
One easy integration of $\sec x$ substitutes $u=\sin x$, viz.$$\int\frac{\cos x}{1-\sin^2 x}\,\mathrm{d}x=\frac{1}{2}\ln\left|\frac{1+\sin x}{1-\sin x}\right|+C.$$Multiplying top and bottom by $1+\sin ...
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How to introduce trigonometric ratios (HS) through a cognitive model?
I'm teaching a precalculus course and also taking a class on how to teach mathematics constructing a specific cognitive model for different topics. So, I have this assignment to build a cognitive ...
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Why teach reference angles?
Reference angles are most useful for limited trigonometric tables with values from 0 to $\frac{\pi}{2}$. They can also help illustrate the periodicity of trigonometric functions, but I feel this is ...
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May we permit identities to be established by equivalent equations?
A trigonometry text like Sullivan's Algebra & Trigonometry often has a prohibition like this (Sec. 7.3):
WARNING: Be careful not to handle identities to be established as if they were ...
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Trig Tables and Right Triangles
I realize that trig tables are somewhat out-of-fashion, but I think my question still makes sense in the computer age: why do trig tables treat the case of a right triangle instead of an arbitrary ...
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Memorizing Trig Identities
I adjunct for a local community college teaching College Algebra and College Trigonometry. Every year, the community college math department insists on students memorizing each of the trig identities....
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Applying inverse trigonometric functions
I am teaching a Trigonometry class, and every year we get to this point my students start asking a lot of questions. And for good reason.
Here is my issue: We teach that $\arccos\frac{1}{2}=\frac{\...
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Solving a Polar set of equations algebraically?
I was coaching a student on how to approach this problem. 2 equations given and the question was where they intersect.
Now, with a bit of practice on polar coordinates, producing the graph by hand ...
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How to explain this trigonometric problem properly?
Trigonometry has some complex stuff, but sometimes students have trouble with the beginning, the basics.
As an example, here's what seems to be a simple question:
$$2cos(x)=1$$ $$0°≤x≤360°$$
...
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Resources on 3D transforms, vectors, coordinate systems
Background: I'm helping engineers use software to create 3D geometry in a programmatic way (similar to OpenSCAD). The functions they need to call have inputs which are low-level geometry concepts: 3D ...
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Rules to eliminate erroneous solutions in Trig equations?
This is a High School Trig problem asking for the solution to an otherwise simple equation.
$\frac{\left(1+\cos x\right)}{\sin x}$=-1 (per comment - the domain was specified as greater than -180 ...
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Necessary trigonometric identities for k-12 students [closed]
Which trigonometric identities are necessary for the k-12 student?
I want to make a list of trigonometruc identities for k-12 student. My list is below, please help me to improve the list by ...
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How are the basic trigonometric functions introduced to students?
The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is ...
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Rigorously defining the concept of an angle for high school students
Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected.
Google search provided me with many definition that are too technical or too vague IMO.
...
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Trigonometric angles of rotation
I find that the notion of trigonometric angles of rotation is a bit confusing for the students.
In my curriculum, students learn about angles first time at geometry in middle-school. The angles are "...
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Explaining trigonometric equations
I want to know what is the best way to teach simple trigonometric equations, such as $$\sin x=0$$.
Should I use the trigonometric circle or the sine graph? Which is better?
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How to convince my student that this is an Identity : $\sec^2x-\tan^2x=1$?
I taught three basic trigonometric identities. One of my 9th grade students asked: How can we say $$\sec^2x-\tan^2x=1$$ is an identity since when we plug in $x=\frac{\pi}{2}$ the identity fails which ...
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Why do we conventionally treat trig functions as going anti-clockwise from the right?
I realise that teachers tend to focus on right-angled triangles when introducing trig functions, and for those I can see that the most intuitive approach seems to be starting with the opposite and ...
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What purpose do these kinds of question serve in mathematical training?
The question in the picture is a relatively easy one to answer. But while learning Trigonometry, I had to answer countless questions asking to prove certain relations between obscure and complicated ...
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Examples of Mathematical Beauty in School Mathematics
Various branches of mathematics have mathematical beauty. Some of this are visual, such as the mandelbrot set, while others are logically sublime, such as the recursive simplicities of peano ...
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Should word problems be reasonable?
I've recently run across a series of problems that didn't reflect reality.
For example -
An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH.
...
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How to convince the students of grade 8 that $\sin 90^\circ =1$? ( calculator not allowed )
In general the secondary student may not ask why is $\sin 90^\circ = 1$ because they can see the answer in the graph of the sine wave. However the students in grade 8 are not familiar with the graph ...
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Examples of when $\tan(x) = \frac{\sin(x)}{\cos(x)}$ is useful
I'm making a video right now about the unit circle definitions of the basic trig functions.
I've done sine and cosine, and am now talking about the tangent function.
As most of you know, it can be ...
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The origins of $\operatorname{cis}(\theta)$
There is a abbreviation used in high school mathematics that is almost never seen outside of it: $\operatorname{cis}(\theta) = \cos(\theta) + i \sin(\theta)$, where cis stands for cosine + i sine.
As ...
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The values of trigonometric ratios
Today I explained to my students how the value of $\sin(90°)$ is 1 by first drawing a right angled triangle and taking $90°$ as a reference angle, and told them that now $\sin(\theta)=\frac{\text{...
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How to motivate the geometric definition of trigonometric functions on the unit circle
Suppose your students know already the geometric definition of $\sin$ $\cos$ and $\tan$ for angles between $0^{\circ}$ and $90^{\circ}$.
How can I motivate the definition of $\sin$, $\cos$ and $\tan$...
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Proving trigonometric identities
When teaching trigonometric identities, I found the students had trouble proving them. All the students taking turns to ask almost all the questions of an exercise embarrased me somewhat. In order to ...
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Trigonometric ratios
How should I teach students the trigonometric ratios who are indeed studying trigonometry for the first time in their life? The course has said that there are six ratios based on sides in a triangle ...
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