Hello Brilliantitians, I've written an article called "Learn Trigonometry in One Day," which I posted here. Share this with anyone you think may want to learn trigonometry! :)

I read through the whole thing [except for the 'solutions to the exercises' part]

Here's what I think.

1) It's concise, brief and to-the-point. This can be both good and bad. It would be unfair to assume that someone can learn trigonometry completely from scratch by reading your article. A person needs to have a little background on trigonometry before they can read your article.

2) The way you defined sines and cosines of angles in chapter 1 makes sines and cosines only non-negative. So exercise 1.1 should not ask the reader to prove that \(\sin\theta\) and \(\cos\theta\) are never less than \(-1\).

3) In chapter 2, I'm not sure what you mean by radians are unitless. The radian itself is a unit. It is the unit of angular measure. And the radian does have a symbol, the superscript c [\(2\pi^{c}\)]. But people do not use it as it may give rise to confusion.

4) I'm not totally sure how you do it in the USA, but here we learn complex numbers way after we learn basic trig. Using Euler's formula to derive trig identities is probably not the best way to teach people who are learning trig for the first time.

5) Other than that, I think it's quite well-written with enough diagrams and those are really important while learning trig.

I see that you have other articles on academia. I'm looking forward to reading them later today.

@Mursalin Habib
–
Radians are unitless... we know that \(s=r\theta\) where \(s\) is arc length, \(r\) is radius, and \(\theta\) is angle in radians. Therefore, \(\theta\) is unitless.

As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted.

@Cody Johnson
–
Notice that I didn't disagree with anything you said.

\(s=\frac{\pi}{180}r\theta\) where \(\theta\) is in degrees. Therefore \(\theta\) is unitless?

Neither \(\theta\) is unitless. Both \(\theta\)'s have a dimensionless unit.

To say radians are unitless is the same as saying that meters are unitless. Both meters and radians measure a measurable physical quantity. Meters measure length, displacement; radians measure angles, angular displacement. And that's what units do, don't they? They measure stuff.

The only difference between the meter and the radian is one of them has dimensions while the other one doesn't.

It can also be argued that "pure numbers" have units. \(1\) is the unit for positive integers. \(\pm 1\), \(\pm i\) are the units for Gaussian integers.

As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol

I agree. It needs no unit symbol. But does not mean that it is not a unit. A degree is also the ratio of two lengths times a constant.

It all boils down to what you mean by a unit. To me a unit is a standard amount of a physical quantity. A radian is specifically that. It just happens that the physical quantity it represents has no dimensions.

I didn't want to comment on this anymore. But I wanted to see how the "backfire effect" worked and this seems like a perfect illustration of that.

It's ironic how I taught a few trigonometry and complex number classes to middle and high school students at MIT last week (taught it 3 times). These seminars were one 1 hour long, each. (Not a series, separate classes) and the topics I covered and the approach I used was almost the same as those you cover in your article. Nice work! Do you mind if I use this as a handout next time I teach this class?

On page 6, the area formula of triangle \(A\) can be confused with the angle \(A\). You can use \( [ \triangle ABC ]\) instead. Other than that is amazing. ^__^

I'll nitpick a bit, but you wrote, "Similarly, \(\sin\theta >0\) for \(0 <\theta < 180^{\circ}\) and \(180^{\circ}<\theta < 360^{\circ}\)." You omitted "\(\sin\theta<0\) for" after the word and. You also omitted the degree sign above the \(0\), i.e. it is \(0^{\circ} <\theta < 180^{\circ}\).

A good article but a bit concise for beginners who actually learn from scratch. Asking the proof for extended sine rule as a problem was a really good exercise. Kudos!! Looking forward to advanced topics :)

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## Comments

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TopNewestNice job with this. But the area formula isn't \(A = \dfrac12 ab \cos C,\) it's \(A = \dfrac12 ab \sin C.\)

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Oh crap, how could I make such a stupid typo!

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Ha ha yeah i saw it too.

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I read through the whole thing [except for the 'solutions to the exercises' part]

Here's what I think.

1) It's concise, brief and to-the-point. This can be both good and bad. It would be unfair to assume that someone can learn trigonometry completely from scratch by reading your article. A person needs to have a little background on trigonometry before they can read your article.

2) The way you defined sines and cosines of angles in chapter 1 makes sines and cosines only non-negative. So exercise 1.1 should not ask the reader to prove that \(\sin\theta\) and \(\cos\theta\) are never less than \(-1\).

3) In chapter 2, I'm not sure what you mean by radians are unitless. The radian itself is a unit. It is the unit of angular measure. And the radian does have a symbol, the superscript c [\(2\pi^{c}\)]. But people do not use it as it may give rise to confusion.

4) I'm not totally sure how you do it in the USA, but here we learn complex numbers way after we learn basic trig. Using Euler's formula to derive trig identities is probably not the best way to teach people who are learning trig for the first time.

5) Other than that, I think it's quite well-written with enough diagrams and those are really important while learning trig.

I see that you have other articles on academia. I'm looking forward to reading them later today.

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I didn't say that they weren't. But the radian is an SI-derived unit. It's dimensionless.

Here's Wikipedia to back me up.

Imgur

The main issue I had was the sentence "radians are unitless" since they are units that measure angles.

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On your very same Wikipedia page,

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\(s=\frac{\pi}{180}r\theta\) where \(\theta\) is in degrees. Therefore \(\theta\) is unitless?

Neither \(\theta\) is unitless. Both \(\theta\)'s have a

dimensionlessunit.To say radians are unitless is the same as saying that meters are unitless. Both meters and radians measure a measurable physical quantity. Meters measure length, displacement; radians measure angles, angular displacement. And that's what units do, don't they? They measure stuff.

The only difference between the meter and the radian is one of them has dimensions while the other one doesn't.

It can also be argued that "pure numbers" have units. \(1\) is the unit for positive integers. \(\pm 1\), \(\pm i\) are the units for Gaussian integers.

I agree. It needs no unit

symbol. But does not mean that it is not a unit. A degree is also the ratio of two lengths times a constant.It all boils down to what you mean by a unit. To me a unit is a standard amount of a physical quantity. A radian is specifically that. It just happens that the physical quantity it represents has no dimensions.

I didn't want to comment on this anymore. But I wanted to see how the "backfire effect" worked and this seems like a perfect illustration of that.

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GJ CJ !

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Awesome article! I am new to trigonometry and it has helped me a lot. Thank you for taking the time to write this. :)

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It's ironic how I taught a few trigonometry and complex number classes to middle and high school students at MIT last week (taught it 3 times). These seminars were one 1 hour long, each. (Not a series, separate classes) and the topics I covered and the approach I used was almost the same as those you cover in your article. Nice work! Do you mind if I use this as a handout next time I teach this class?

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Sure!

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Thanks!

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@Cody Johnson @Ahaan Rungta Can you add these notes to the Trigonometry section on the Wiki? Thanks!

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Great work Cody!

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it is comprehensive for beginners

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Where is the article? I'd like tor read it, but I can't find it.

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On page 6, the area formula of triangle \(A\) can be confused with the angle \(A\). You can use \( [ \triangle ABC ]\) instead. Other than that is amazing. ^__^

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Amazing well done and explanatory really helped thanks

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Thank you all for your feedback. I will edit and update the article.

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Thanks Cody for the trig lesson.

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I'll nitpick a bit, but you wrote, "Similarly, \(\sin\theta >0\) for \(0 <\theta < 180^{\circ}\) and \(180^{\circ}<\theta < 360^{\circ}\)." You omitted "

\(\sin\theta<0\) for" after the wordand. You also omitted the degree sign above the \(0\), i.e. it is \(0^{\circ} <\theta < 180^{\circ}\).Log in to reply

Haha, I fixed it.

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I like the article but it can't teach you trigonometry in one day.

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A good article but a bit concise for beginners who actually learn from scratch. Asking the proof for extended sine rule as a problem was a really good exercise. Kudos!! Looking forward to advanced topics :)

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