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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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2
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2answers
125 views

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 ...
6
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1answer
124 views

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. ...
15
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9answers
4k views

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 ...
6
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2answers
254 views

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 ...
9
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4answers
174 views

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 ...
4
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4answers
316 views

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 ...
1
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1answer
111 views

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 ...
7
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3answers
342 views

“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 ...
5
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4answers
341 views

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 ...
9
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4answers
492 views

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 \...
17
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4answers
4k views

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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2answers
406 views

Why do we state the antiderivative of $\sec x$ as $\ln |\sec x +\tan x|$?

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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0answers
113 views

Is there a simple rationale for learning “reference angle” I can give a 9th grader?

I'm helping a 9th grader review for his Algebra 2 Regents exam (New York State). They need to know how to find the "reference angle." (I did read Why teach reference angles?.) I haven't found a ...
1
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0answers
57 views

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 ...
9
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3answers
1k views

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 ...
8
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4answers
660 views

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 ...
2
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1answer
245 views

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 ...
17
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8answers
3k views

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....
8
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4answers
261 views

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{\...
-1
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4answers
120 views

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 ...
2
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3answers
374 views

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: $$2cos(x)=1$$ $$0°≤x≤360°$$ ...
3
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1answer
97 views

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 ...
7
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3answers
343 views

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 ...
4
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2answers
179 views

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 ...
4
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3answers
1k views

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 ...
11
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5answers
1k views

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. ...
6
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1answer
1k views

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 "...
2
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3answers
261 views

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?
17
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9answers
2k views

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 ...
17
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12answers
3k views

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 ...
5
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2answers
191 views

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 ...
10
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4answers
700 views

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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5answers
2k views

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. ...
8
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5answers
351 views

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 ...
8
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3answers
354 views

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 ...
11
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1answer
194 views

The origins of $\mathrm{cis}(\theta)$

There is a abbreviation used in high school mathematics that is almost never seen outside of it: $\mathrm{cis}(\theta) = \cos(\theta) + i \sin(\theta)$, where cis stands for cosine + i sine. As soon ...
5
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3answers
233 views

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{...
9
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5answers
5k views

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 $\tan$...
5
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5answers
2k views

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 ...
6
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1answer
141 views

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 ...
8
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2answers
742 views

Unit circle vs. ratios of right triangles vs. wave functions for introducing trig functions

In an introductory trignometry course, there are many options for introducing trigonometric functions: As ratios of sides of right triangles As coordinates (or ratios of coordinates) of intersections ...
14
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8answers
2k views

Why are degree angle-measurements taught?

Apologies if this question has some obvious answer. Why are degrees still taught as a measure for angles, to be replaced later by radians (probably confusing many people), rather than just starting ...
26
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4answers
2k views

The best way to introduce trigonometric functions in a rigorous analysis course

This is something I have always had issues with. Generally, three approaches are used: The geometric path: this follows the standard way how you would introduce these functions in school. The problem ...
21
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10answers
4k views

Pi or Tau? How should the circle constant be taught?

Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent ...
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6answers
2k views

Answers in exact form (e.g. including radicals) vs. Decimal Approximations

I was tutoring a student on early trigonometry. Solving for the hypotenuse of a right triangle, but with sine, not Pythagoras. The student went through getting the sine of a 45° triangle, and gave ...