determine the angles of the rt triangle.?

Question

read v as root over.
base as v3, perpendicular as 1 and hypotenuse as 2.
any given number having any given power can be expressed as difference of infinite sets of two perfect squares.prove it by number theory.

Answer 1

(√3)² + 1 ² = 2²
sin α = √3/2 α = 60º; ß = 90 - α; ß = 30º

And your second question is
aⁿ = c² - b². I'm going to assume that you wanted to put in a, b and c are all integers and a > 1. Because if not, your theorem is not true and cannot be proven. With that stipulation, this theorem passes the "intuitive" test. It makes sense.
We know for certain that, for all even n, it must be true by the Pythagorean theorem. So only odd n are of consequence to this theorem.

I'm going to take a moment here and say that if you're trying to use this theorem as a FLT lemma, you must understand that it won't fly. The lemma must use the carry the same stipulations as the successor theorem. Said the other way, the corollary is bound by the restrictions of the theorem.

a=2, n=3 gives us c=3 and b=1. 8=9-1. I took some time trying to find a counter-example and could not. I did find some neat patterns emerging.

The proof to your theorem is simple: aⁿ = a²[(½a+½)² - (½a-½)²] because the expression in brackets will always be equal to a (a is odd prime; composite a can be simplified to prime a).

I also found that 2ⁿ= [2^(n-2)+1]² - [2^(n-2)-1]². So, in addition to all the solutions from the above expression, there will be another set of solutions for 2ⁿ; n is odd integer > 5.

Answer 2

First, "any given number having any given power" is meaningless. U can give any power to any number. You are trying to say all numbers, and its powers. But this is unnecessary as well as all powers of numbers r themselves numbers too.

"difference of infinite sets of two perfect squares". difference of sets?? a set A={1,2,3,4,5,6,7} and a set B={1,2,4,5,6,8}.... the difference of these 2 sets is {3,7,8}

You r using the wrong terminologies and it's a big no no in math as rigorous mathematical language is created as a universal language to describe precise notions and if u go around misusing them, u r reducing mathematics into just another ordinary language.

From what u say in your other questions, I think all u r trying to say is, "Any number N can be expressed as a difference of 2 squares: N = a^2 - b^2"

See, this is so simple. why do u want to confuse everyone with geeky but flawed language?

and to prove N = a^2 - b^2,

N = a^2 - b^2 = (a+b)(a-b) = c.d

Any number N can be expressed as the product of 2 numbers c and d. This is obvious enough.

The trivial case is when c and d are 1 and N. since u have 2 unknowns and 2 equations a+b=c and a-b=d, a and b can be found.

This applies not just to integer N, a and b but for all number system.

Answer 3

That is a 30-60-90 triangle. The arctan is equal to the side opposite the desired angle divided by the side adjacent to that angle so for the smaller of the two angles arctan1/v3 or v3/3 so your calculator doesn't say error, that comes out to 30 degrees so the other angle is 60 degrees.

Answer 4

a triangle is 180 degrees the 2 sides of the right triangle is 90. subtract that from 180 and thats your answer.