Help me

Getting the exact values from WolframAlpha (knowing I could have worked them out myself if I had the time), I find

sin(12) = 1/4 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))​
sin(48) = 1/4 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))​
sin(18) = 1/4 (sqrt(5) - 1)​
sin(30) = 1/2, of course​

Given these, it's not hard to show that (sin(12)sin(48))^2 = (sin(18)sin(30))^2 = (3 - sqrt(5))/32. This is equivalent to the equation we got from the law of sines. So we've proved that the exact answer is 12.

As to whether one could ever figure this out without first "guessing" that the answer is exactly 12 degrees, I have no idea!

I still want to see the exact wording of the original problem.
 
I am very adventurous. No one was found on this forum who will solve this problem ??

Maybe there is another way to solve this task and find it all together
 
tato1982 said:
I am very adventurous. No one was found on this forum who will solve this problem ??
Click to expand...
I don't see any lack of trying! Some tasks are just very difficult, and it is inappropriate to blame people for not solving them. It takes more than adventurousness. It may even take more than genius.

But you still haven't given us a reason to think it should be reasonably easy. What does the original problem say, as given to you?
 
I could not understand what to say. We could not solve the task until the end. Maybe there is another way to draw something in a triangle or a bisector, or a height, or a parallel line. Let's think about it together.

I know we are not geniuses and it is a difficult task really.I hope someone solves this task to the end.Thanks for standing by you are good people on this forum
 
tato1982 said:
∆SDT
ST/sin<sdt=TD/sin48
∆TDV
TD/sin18=DV/sinx

ST/TD=sin48/sin<std
DV/TD=sinx/sin18

ST=TD
sin<std= x+18

sin48/sin(x+18)=sinx/sin18

find trigonometry?????
Click to expand...
Exactly what are you trying to find?
 
There is a similar thread here:
mathhelpforum.com

Trig Problem

I got that too but in the end I could not solve the equation Do you mean solving for x without using a calculator? I hadn't intended to be so brave! Someone smart could probably do that. I had imagined solving \cos 18 + \cot x \sin 18 = \dfrac{\sin 48}{\sin 18}\\ \cot x \sin 18 =...
mathhelpforum.com mathhelpforum.com
 
I made a few observations which I'll share for anyone else who finds this thread interesting. I wanted to see what happens with some other scenarios. To me it did seem a bit surprising to obtain an exact integer angle (in degrees) from this problem. So, let ∠TSD = y, ∠TVD = z and then...

tan(x)=sin(z)^2 / (sin(y) - sin(z)*cos(z))

In the original image y=48 and z=18. I plugged in lots of other integer values for y and z and looked for integer values of x from the above equation. This revealed the following probable relationships...

if z=y then x=90-z/2 for example (x,y,z) can be (89,2,2), (88.5,3,3), (88,4,4), (46,88,88)...
if z*2=y then x=z for example (x,y,z) can be (1,2,1), (2,4,2), (3,6,3), (44,88,44)...
if 180-y=z*2 then x=z for example (x,y,z) can be (46,88,46), (47,86,47), (48,84,48), (89,2,89)...

The above conjectures generate many more integer sets of solutions (and also infinite real valued solutions). I proved one of them using trig. You may be able to visualize why they work.

The following probable integer values also appeared which don't fit into the above categories...
Code: 
 x   y   z  
==========
12  48  18   (this original scenario has been proved to be exact)
10  80  20  
50  40  30  
24  84  30

I have no intuitive explanation for these, but I didn't spend a lot of time researching them.
 
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