There are many right triangles whose area has as few prime factors as possible:

The Green-Tao paper "Linear equations in primes", and the subsequent work on their Mobius Nilsequences conjecture by GT and Gowers Inverse conjecture by GT-Ziegler, implies that $\frac{ab}{12}$ is infintely often a product of four primes, in a quantitative sense.  Indeed, reparametrizing gives $ab=12xy(2x+3y)(2x-3y)$, and now we are asking for points in the lattice $(1,0,2,2)\mathbb{Z}+(0,1,3,-3)\mathbb{Z}$ all of whose coordinates are prime.  This system has finite complexity and thus the main results of "Linear equations in primes" applies unconditionally, that is to say the number of pairs $(x,y)$ with $0 &lt; |x| , |y| &lt; T$ and $xy(2x+3y)(2x-3y)$ a product of four primes is asymptotically $cT^2 (\log{T})^{-4}$ for some constant $c$.  

This deduction can be found, for example, in Sarnak's notes "Equidistribution and primes".

Does the function which sends a right angled triangle to its area produce infinitely many numbers having hardly any prime factors?

  Use sketch to find exact value of $\tan (\cos^{-1}\dfrac{5}{13})$


I drew a right triangle with angle $\theta$ and sides $12,5,3.$
If $\cos \theta=\frac{5}{13},$ then $\sin \theta = \frac{12}{13}$ and $\tan \theta = \frac{12}{5}.$  

This isn't correct since tangent is greater than one. How would I solve this correctly? (Please show steps) Thanks.
