Forum Thread

In a geometry project, I have chosen the option of completing a difficult problem. This problem, which can be found online when searching up "hard geometry problems", is actually hard to do, but the solution can be found in many places. However, the "easy" solution involves many auxiliary lines and if I were to present that as a proof, it would be obvious that I had searched it up. Instead, I decided to use trigonometric equations (which are past the level of the class) to solve for the angle measurement that is needed. I got to a certain point, but eventually, I just had to use my calculator to finish the problem, getting the (according to the web) correct answer. After this, I went back and tried for a long time to solve the final step by hand, but could never get all the way to the answer, and finally, I decided that it would be better to ask a more experienced person to show me how to do this. Technically, this wouldn't really be cheating; this method is past the level of content that my (high school honors geometry) class is already teaching, and my teacher would accept the answer if I were to just write that I had used my calculator. I do know the trigonometric identities relatively well, but wasn't able to figure out what to do after a certain point. Here is the relatively simplified version of the equation:

(sin(80)-2sin(40)+tan(60))/(2sin^2(40))=cot(x)

(Note: this is obviously in degrees.)

I'm sure that this will be extremely easy for some of you, and I am willing to pay a small PH prize to the first person who gives a viable and thorough explanation of how to reach the answer. By the way, if it helps, the calculator says that x = 30 degrees.

Prizes (pick a few):
A dragon gem
2 brown boxes
A green, pink, and red key
Spray Duck
Water Stone
A bunch of other gems

Or if you want something better, I can try to provide something else.

Thank you!

I'm no good at math so I can't help you. But tag your title with the bracket thingies.
p.s. I like your avatar.

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Like iammil already said, please tag your thread or else it'll be deleted soon.

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Thanks for reminding me; I didn't realize I had to do that. And thanks about the avatar; I bought it from someone on Pokefarm (1).

BTW I still have another 10 days to do this before I just have to say that I used a calculator, so don't feel rushed.

(sin(80)-2sin(40)+tan(60))/(2sin^2(40))=cot(x)

First, we know that all of these terms except cot(x) will have a specific decimal value. We're going to convert all of those terms into decimals, which may be easier to deal with.
The only real way to get these values is by using a calculator. I will be going seven decimal places.
sin(80) = 0.9848078
2sin(40) = 1.2855752
tan(60) = 1.7320508
2sin^2(40) = 0.8263518

Now we can plug the decimal values back into the original problem.
(0.9848078 - 1.2855752 + 1.7320508)/0.8263518 = cot(x)

Adding/subtracting what is in parentheses will get us...

1.4312834/0.8263518 = cot(x)

Then you can get back to the trig identities. We know that cot(x) is equal to 1/tan(x)

1.4312834/0.8263518 = 1/tan(x)

Now we're looking to get tan(x) all by itself so that we can use arctan later. Multiply both sides by tan(x)...

1.4312834/0.8263518 * tan(x) = 1

Then you can divide both sides by 1.4312834/0.8263518...

tan(x) = 1/(1.4312834/0.8263518)

Good, now tan(x) is by itself. We can simplify the other side since 1 over a fraction always gives the reciprocal...

tan(x) = 0.8263518/1.4312834

Good deal, this is getting simple now! Divide...

tan(x) = 0.5773502

Alright, now, we're at the final stretch. We want to get x all by itself, and we can do that using arctan.

x = arctan(0.5773502)

A calculator quickly reduces this to...

x = pi/6 radians
which is the same as
x = 30 degrees.

Hope this helped! :)

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Dangit.
Lol, I was about to explain it in a totally different way (applying arccot(x) to both sides and etc.) but you solution seems neat.

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Thanks so much! Is there any specific bonus that you want?

I would love that Dragon gem :P
I also noticed you have a Tyrantrum in one of your boxes... I was wondering if you could possibly breed me one? If not, the gem is more than enough <3

Thank you!

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Lol, I used it in the cauldron, sorry XD. (I got a mega stone though)

You can have the Tyrantrum if you want it.

Oh, that's fine! I'd love the Tyrantrum :P

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It's in the GTS with your name on it. Enjoy!