Final answer:
Trigonometry allowed for accurate computations with large numbers in ancient mathematics due to its consistent geometric relationships and logical structure, even before the advent of calculators. Ancient mathematicians like Archimedes achieved high accuracy with a disciplined approach to precision and estimation.
Step-by-step explanation:
Trigonometry allows for some questions to be answered very accurately even with large numbers due to the nature of its logical structure. Each trigonometric calculation is based on a set of postulates, meaning that for any given inputs, you will always arrive at the same consistent outcome regardless of the size of the numbers. Ancient mathematicians like Archimedes achieved high precision using geometric methods and trigonometry with an accuracy comparable to about four decimal places. Calculators and significant figures play a role in modern computation, highlighting a balance between the need for precision and practicality.
However, trigonometry was particularly helpful because its principles are rooted in geometric relationships rather than just numerical calculations. Since these relationships are consistent, they can yield accurate results without dependence on the large numerical accuracy that calculators provide. Moreover, because ancient mathematicians recognized that certain numbers could not be precisely expressed, they often relied on estimates with a known level of precision, similar to how we use significant figures today to decide the level of precision necessary for a calculation.
Therefore, trigonometry provided a reliable framework for achieving accurate results even when computations involved large numbers or when approximations were necessary due to the impracticality of exact calculations.