SOLUTION: A right triangle has a hypotenuse of length 37, and the lengths of both of its legs are integers. Find the sum of the sine of the smallest angle and the secant of the smallest angl

Question 1029872: A right triangle has a hypotenuse of length 37, and the lengths of both of its legs are integers. Find the sum of the sine of the smallest angle and the secant of the smallest angle of this right triangle. Express your answer as an improper fraction reduced to lowest terms.
Answer: 1789/1295
There must be some law that could explain this for me. Could you please help??
Thank you so much!!
Found 2 solutions by Fombitz, rothauserc:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
So in other words, the sum of the squares of the legs has to equal 37 using the Pythagorean theorem.
There's only one pair, and , since .
So then, the smallest angle of this triangle occurs where,

So then,

and the sum is,

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
We are given that the hypotenuse of a right triangle is 37 and
:
a^2 + b^2 = 37^2 = 1369
:
37 is a prime of the form (4k+1) and 1369's prime factorization is 37^2
:
If a whole number's prime factorization has primes only of the form (4k+1), then it can be written as the sum of two squares
:
From the definition of a triangle, we know that
:
a + b > 37
:
Consider perfect squares less than 1369
:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289,
324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961,
1024, 1089, 1156, 1225, 1296
:
12 and 35 are the only pair of integers that satisfy the following
:
12^2 + 35^2 = 1369
:

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