Is Trigonometry or the Pythagorean Theorem more accurate?

My geometry teacher is away on maternity leave and I have this long time substitute, and she is really making me mad! It always seems like she has no idea what she is doing, and she is VERY picky when grading! If you don't have EXACTLY what she is looking for, you will get less points! We are learning about Area of regular polygons in relation to circles, and I feel more comfortable using the pythagorean thereom to find the apothem (height of equilateral triangles that make up a regular polygon). It's simple: the hypotenuse is one side length, the short leg is half that, and you use the Pythagorean theorem to find the missing leg! Now she is claiming that we are supposed to find it using trigonometry, but I seem to believe that trig isn't as accurate! Which is more accurate? I want to be able to prove her wrong, so please explain thouroughly! Plus I found out that calculators find trig based on estimating it from information stored by the Taylor Series.