When I studied in class ten, I used to think if the measures of the angles of a right angled triangle could be obtainable without using trigonometric tables or scientific calculators?
It took me a while to formulate the formula. I amazed when the observed values came close to the calculated values and variation in values occurred by 0-0.5.
Formulation:- If ΔABC is a right angled triangle where m∠B=90° and AB=c, BC=a & AC=b units (AB
m∠A= (?×?)÷? (Locked) m∠C= (?×?)÷? (Locked)
Observation:-
1). 3,4 & 5 units 36.87°, 53.13° & 90° (Calculator) 36.5°, 53.5° & 90° (Formula) Variation =36.87-36.5=0.37
2). 5,12 & 13 units 22.62°, 67.38° & 90° (Calculator) 23°, 67° & 90° (Formula) Variation=23-22.62=0.38
3). 7,24 & 25 units 16.26°,73.74° & 90° (Calculator) 16.68°, 73.32° & 90° (Formula) Variation=16.68-16.26=0.42
The numerical approximations to the answers can be obtainable.
Of course you can. You can do any computation your computer can do if you have enough energy and patience. People calculated $\pi$ to hundreds of places before the age of calculators and computers. How much accuracy do you want and how much work are you willing to do? The small angles are easier. You can use the Taylor series $$\arcsin x=x + x^3/6 + (3 x^5)/40 + (5 x^7)/112 + (35 x^9)/1152 + (63 x^{11})/2816 + O(x^{12})$$ For the $7-24-25$ triangle $x=0.08$ so this many terms should get you fifteen places or so. For the $3-4-5$ the angle is larger and you would probably want to expand around $\frac \pi 4$ or to use the half angle formulas to get a smaller angle