As everyone knows the pythagoran theorem can be used on triangles with 90 degrees angles using "2" as an exponent. Then on my geometry class I learned that for 3 numbers in order to be able to form a triangle they have to comply this " c < a+b "
I figured out that in an obtuse triangle pythagoras should work with different exponents between 1 or 2. I created a python program to realize which exponents to use in each case. For example for a 120 degree triangle the program showed that I had to use the exponent 1.26. Pythagoras Theorem works if I substitute the exponents by 1.26. "c^1.26 = a^1.26+b^1.26". Pythagoras with these exponents works in this case. I know about the law of cosines and the law of sines, but I think if there was only one way to use pythagoras with alternative exponents would be much neat.
I found the exponents that will fit to the pythagoran theorem in each case where I increase or decrease the angle between the values 179 to 61. Then I plotted a graph in excel relating the angle with the exponent that I had to use in each case, and I got this nice looking graph
I tried to make a trendline with help of excel, I tried polynomial, exponential, but none of those trendlines seemed to fit. At this point I have a table to know which exponents to use in each case, but I think a formula obtained from the graph would be much helpful.
How can I obtain a formula from this graphs?
Your method doesn't work. Suppose the angle between sides $a$ and $b$ is $120°$, then the solution of $a^x+b^x=c^x$ is:
$x = 1.26186$, for $a=1$ and $b=1$;
$x = 1.25250$, for $a=1$ and $b=2$;
$x = 1.24001$, for $a=1$ and $b=3$;
$x = 1.25855$, for $a=2$ and $b=3$;
and so on: for different ratios $a/b$ you get different values of the exponent.
If $a=b$ then $c=2a\sin{\theta\over2}$ and equation $a^x+b^x=c^x$ becomes $2a^x=\big(2a\sin{\theta\over2}\big)^x$, which can be solved to give: $$ x={\ln2\over\ln\big(2\sin{\theta\over2}\big)}. $$ This is then the equation of your "nice looking" graph.