My attempt to find the law of tangents gave me a new trigonometric equation — I guess

Laws of Sines and Cosines, but where is the Law of Tangents?

Yes, I’m a math nerd.

And one of my passions is deriving formulas.

Therefore, after I derived the sine and cosine laws (the sine law was surprisingly easy to come up with, but the cosine law... I spent hours on it.)

I wondered, “Where is the tangent law? We’ve never learned that at school?”

Then, after reflection, I knew that there was no need to teach the tangent law at school, for with the sine and cosine laws, you can effortlessly figure out all the measures of any triangle.

Nevertheless, I gave it a try:

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Please don’t mind the symbol used; I’m just tired of the a,b,c, and d stuff. Each angle carries the same symbol as its facing side (side A → angle A: facing each other)

1 — Simple tangent equations:

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2-Now the sum of the two small sides = a side of the triangle:

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Notice that the tangent of the angles can be used to find the measure of a side. Now the goal of the game is to find a way to rewrite that symbol in both terms in terms of the triangle’s sides’ symbols, which was quite a brainy task.

After hours of work and tons of piles of paper, I found the method.

1. Factorise (that becomes this) :

2. Spot the right triangle: (this stuff)²+(that stuff)² = (another stuff)²

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3. Use the old equation to make a new one.

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4. Isolate

You can’t imagine how long and attention-demanding this part was...

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5. Substitute

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6. Create a variable for clarity

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7. Expand, cancel out, again and again, until finally…

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This long equation is just equal to zero

Can you spot the hidden equation?

It took me quite some time to spot it, and when I had the revelation I…

8. Use a math trick

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9. Do you love quadratic equations like me?

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10. Factorise and Reduce

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11. Remember

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I know the equation is long…

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But after expansion and cancelling out, the equation is…still long

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11. Try it out

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These are measures taken with my instruments

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After computation, I get

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It works!

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With your a, b, c… the expression looks like this:

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→Finallyᕕ( ཀ ʖ̯ ཀ)ᕗ

As for now I haven’t seen a case where the ( — ) gives the real answer

The real law of tan

After figuring this out, I checked the real law of tan.

You can imagine how shocked I was to to see this:

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What, why, how ?!

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They don’t look alike at ALL.

After a moment, I thought if I continued reducing the equation, I would eventually arrive at the real law of tan, but…I’m lazy, figuring out my law of tan took me hours, and honestly, I have no idea how to continue. Therefore, if someone is clever enough to continue:

Let him enjoy himself

Thus, share this article with anyone out there who enjoys math.

Thank you for reading! 😊 Have any queries or ideas? Leave them in the comments section 💬 ; I look forward to hearing from you! 🤗