Through the usage of trigonometry and a protractor.
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There is no single answer because it all depends on what you know about the triangle and also what you need to find out. In general, trigonometry is useful.
a way to analyze and solve problems using the units, or dimensions, of the measurements.
No, only your brain can. Similar triangles can be used to solve some problems but not others but it is for you to work out - using your brain - whether or not similar triangles are relevant and then to figure out how they might be used.
Using a protractor is the simplest way of solving the measurements of angles.
by experimenting..
Calculus was invented to solve physics problems, so the importance of studying calculus is to solve physics problems.