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#### sweatingbear

##### Member

- May 3, 2013

- 91

I received an excellent question about significant figures when there is no actual numerical data given. In the figure below, we have a cube circumscribing a triangle and the task is to find the angle at M. N and M are midpoints of the cube's side.

By using the Pythagorean theorem a few times and ultimately law of cosines, one can arrive at

\(\displaystyle \cos (\text{M}) = \frac { \frac 94 - \frac 12 - \frac 54 }{ -\sqrt{\frac 52} } \, ,\)

where one solution is \(\displaystyle M \approx 108.4349488^\circ\). Here is the problem: How many significant figures ought one to have for the angle? We were not given any numerical data, so what ought one to do? Integral values seems to be a natural, conventional practice; however, in terms of significant figures, the questions becomes much more interesting.

By using the Pythagorean theorem a few times and ultimately law of cosines, one can arrive at

\(\displaystyle \cos (\text{M}) = \frac { \frac 94 - \frac 12 - \frac 54 }{ -\sqrt{\frac 52} } \, ,\)

where one solution is \(\displaystyle M \approx 108.4349488^\circ\). Here is the problem: How many significant figures ought one to have for the angle? We were not given any numerical data, so what ought one to do? Integral values seems to be a natural, conventional practice; however, in terms of significant figures, the questions becomes much more interesting.

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