Using an isosceles triangle and basic trigonometric identities to prove the trigonometric double angle formulas

First of all, we construct an Isosceles Triangle like this:

With \(AB=BC=a\) (say). And \(B=2x\). Then \(AD \perp BC\) is constructed.

Subsequently, since the angles \(BAC\) and \(ACB\) are equal, angle \(ACB = 90 - x\), then angle \(CAD=x\).

From triangle \(ADC\):

\(\tan \ x\ =\ \frac{CD}{AD}\)

\(\implies \tan \ x\ =\ \frac{a-a\cos 2x}{a\sin 2x}\)

\(\implies \tan \ x\ =\ \frac{1-\cos 2x}{\sin 2x}\)

\(\implies \cos 2x=1-\sin 2x\tan x\)

Squaring both sides,

\(\implies \cos ^22x=1-2\sin 2x\tan x+\sin ^22x\tan ^2x\)

\(\implies 1-\sin ^22x=1-2\sin 2x\tan x+\sin ^22x\tan ^2x\)

\(\implies \sin ^22x=2\sin 2x\tan x-\sin ^22x\tan ^2x\)

\(\sin 2x\) is not necessarily \(0\), so we can divide both sides by \(\sin 2x\).

\(\implies \sin 2x=2\tan x-\sin 2x\tan ^2x\)

\(\implies \sin 2x\left(1+\tan ^2x\right)=2\tan x\)

\(\implies \boxed{\sin 2x=\frac{2\tan x}{1+\tan ^2x}}\)

Furthermore,

\(\sin 2x=\frac{2\tan x}{1+\tan ^2x}=\frac{2\tan x}{\sec ^2x}=2\tan x\cos ^2x=2\frac{\sin x}{\cos x}\cos ^2x=\boxed{2\sin x\cos x}\)

Now, from \(\tan \ x\ =\ \frac{1-\cos 2x}{\sin 2x}\):

\(\sin 2x=\ \frac{1-\cos 2x}{\tan x}\)

\(\implies \frac{2\tan x}{1+\tan ^2x}=\ \frac{\left(1-\cos 2x\right)}{\tan x}\)

\(\implies \frac{2\tan ^2x}{1+\tan ^2x}=\ 1-\cos 2x\)

\(\implies \cos 2x=1-\frac{2\tan ^2x}{1+\tan ^2x}\)

\(\implies \boxed{\cos 2x=\frac{1-\tan ^2x}{1+\tan ^2x}}\)

From here,

\(\cos 2x=\frac{1-\tan ^2x}{1+\tan ^2x} = \frac{\cos ^2x\left(1-\tan ^2x\right)}{\cos ^2x\left(1+\tan ^2x\right)} = \frac{\cos ^2x-\sin ^2x}{\cos ^2x+\sin ^2x} = \frac{\cos ^2x-\sin ^2x}{1} = \boxed{\cos ^2x-\sin ^2x}\)

Combining the expansions,

\(\tan 2x=\frac{\sin 2x}{\cos 2x}=\frac{\left(\frac{2\tan x}{1+\tan ^2x}\right)}{\left(\frac{1-\tan ^2x}{1+\tan ^2x}\right)} = \boxed{\frac{2\tan x}{1-\tan ^2x}}\)

Note by Arkajyoti Banerjee
1 year, 2 months ago

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All the more reason to abandon CT in favour of RT... creating a demarcation between \(\sin\), \(\cos\) and \(\tan\) actually makes things 100 times harder, especially when it comes to the situation of "double-angles".

Some criticisms:

  1. If you are going to use visuals, could you at least do them to scale? I cannot see any of the markings on the diagram; seeing as you are clearly using GeoGebra, can you at least use LaTeX to do your labelling? You can also increase the size of the font too.

  2. I think you could be clearer in defining "dropping a perpendicular from A"; these terms should be clear to anyone with a basic understanding of geometry.

  3. There are a few places where you will need further clarification as to your steps; in these places, it would be very hard for someone not familiar with the work to follow through what you are doing.

  4. "\(\sin2x\) is not necessarily \(0\)" - What if it is zero? I think you need to rephrase it to "assuming \(\sin2x \neq 0\)", and then explaining what happens if \(\sin2x = 0\).

Gennady Notowidigdo - 6 months, 1 week ago

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  1. Brilliant by default minimizes the size of big images added to a post. If you want to see the actual image then click on it, it'll be enlarged.

  2. If one looks at the diagram properly, he/she should be able to understand that \(D\) is a point on line segment \(BC\) such that \(AD \perp BC\). I don't feel if this is incomprehensible in any context.

  3. Well, guess what? I've shown this exact post to students who have just learnt elementary trigonometry. Almost half of them don't have an affinity towards maths. Still, every one of them was able to understand it and did not complain about its comprehensibility. I really don't know why you both find this to be hard to follow.

  4. Some things are pretty obvious.One must be able to understand and analyze the cases themselves. Moreover, this post focuses on deriving the double angle formulas using elementary trigonometry, so it mustn't be elaborating much on analyzing tiny and obvious cases.

I'm sorry if I happen to sound a little rude, but I couldn't help it as others didn't have problems with this.

Arkajyoti Banerjee - 6 months ago

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Hi Arkajyoti,

Please only think about Gennady as helping you move closer to a more powerful view point.

The key is realizing the power of finite prime fields to explain trigonometry.

I promise that if you explore rational trigonometry... you will have much more powerful tools to help guide students towards not understanding but affinity towards mathematics.

No one understands mathematics. We only pretend to play with the tools of gods.

Lets all pretend and play nicely with these tools, while respecting all voices, even if they are disagreeable.

Actually especially if you find them disagreeable and "rude"... as they may be some of the most powerful and useful ones(at times... not always).

Thanks for you post(as well as Gennady!)

-Peter

Peter Michael - 6 months ago

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  1. That does not excuse using really small fonts for the labels. I shouldn't need to enlarge the image, when it's the calculations and the reasoning that I'm more concerned about.
  2. The diagram is supposed to help one understand the definitions, as opposed to being used to define things. Assuming that one understands what you mean is a BIG mistake, especially in mathematics.
  3. I should probably give examples of what I mean here; early on, when you give your reasoning for the angle \(\angle{ACB}\), you may want to do an extra calculation, to minimise the amount of work on the reader's part. I just pointed out one of about 4-5 things, but I had to be more general in my earlier statement so I would not have to go into many specific examples. To be honest, I think the problem is that you assume the diagram to be the Rosetta's stone in your work and that everything was set in stone after that; if you had been more careful in your definitions, I think you could work around a lot of these issues so that your work is more airtight. But then again, that's just my opinion.
  4. What is obvious to you may not be obvious to m, but in saying that you are somewhat correct. But you are correct. I probably should take my last sentence back, but what I said about assuming \(\sin 2x \neq 0\) instead of what you wrote still stands. I will also add that in more complicated situations, elaboration and tiny details become significantly more important.
If I was you, I would either dismiss the criticism if you find others aren't having problem with this, or acknowledge it and move on. I believe my criticisms have been constructive and clear. I'm assuming something got you so riled up that you had to comment in such a tone. I find it hard to imagine how others would follow this work, especially those I work with...

Gennady Notowidigdo - 6 months ago

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All is to confusing

Annie Li - 1 year, 1 month ago

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I agree. Very poorly written and very hard to follow. I could imagine the average mathematician would struggle to read through this if it was a paper.

Gennady Notowidigdo - 6 months, 1 week ago

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Yeah, it is but an indirect way of deriving the identities using just basic trigonometry.

Arkajyoti Banerjee - 1 year, 1 month ago

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Basic "classical trigonometry" is much too transcendental and full of infinite functions. If we use quadrance and spread and pick a specified field we can say much more interesting things!

Peter Michael - 6 months ago

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@Peter Michael I think the issue here comes down to the fact that either way we will have to surrender to the inevitable fact that the concepts of ratios have no linear aspect to them. It flies against the face of the mathematical "establishment", who try their best to simplify every problem down to linear things. Agreed 100% with the second sentence.

Gennady Notowidigdo - 5 months, 2 weeks ago

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@Gennady Notowidigdo I think surrender is not the right word. This is the beginning of how we learn to truly take advantage of mathematical thought. As an artistic device, as one of creation.

As well... this may not be the mathematical "establishment"''s fault. This may be human nature. You often hear, read, or speak about "keeping things straight forward" I know as a teacher my explanations are supposed to be this way.

But this tautological aspect of mathematics that ratios have no linear aspect to them... Is very much the lynch pin. And students are in for a multiplicity of awakenings in this regard(as well as teachers).

It is actually kind of funny if you think about it. No matter how hard you try to suppress this, you will find that it overpowers all aspects of life as we know it.

Students, teachers, administration, etc. cry and complain and argue if you like... this is simply the truth. And it is where the power lies.

The key to getting people onboard is using finite fields to have the rich number theoretic results begin to find their way into popular culture. The issue is someone needs to begin to have them exposed to the light of day.

Peter Michael - 5 months, 2 weeks ago

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i can not understand anything. explanation anyone??????

Annie Li - 1 year, 2 months ago

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Rational trigonometry will help explain everything.

Peter Michael - 5 months, 3 weeks ago

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Actually, I just can’t be stuffed looking or thinking about the question

Annie Li - 5 months, 3 weeks ago

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@Annie Li Unfortunately, "most people" use sine and cosine to explain simple ideas about triangles. What we want is to use spread and quadrance. This will allow number theoretic results to help point the way towards understanding.

The issue is: we all have a lot to learn about this. So get your popcorn, as the show is just about to get interesting.

Peter Michael - 5 months, 3 weeks ago

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@Peter Michael Nahhhh math is boring

Annie Li - 5 months, 3 weeks ago

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@Annie Li If you think of it as "real" then sure... if you take a more abstract stance about it you will find it has much to offer you. And whats more, you can be greedy about it if you like; self-serving...

Take as much as you can get Annie.

Peter Michael - 5 months, 3 weeks ago

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@Peter Michael Wait what??? Are you talking about being greedy about the popcorn or something?

Annie Li - 5 months, 3 weeks ago

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What did you not understand, the construction of and in the triangle?

Arkajyoti Banerjee - 1 year, 2 months ago

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