A question about significant figures and rounding

[trsnd]

Homework Statement

What is the answer to (72.4meters)*(cos58)? How many significant figures does it have, and why?

Homework Equations

"The least number of significant figures in any number of the problem determines the number of significant figures in the answer."
cos58=0.529919264233...

The Attempt at a Solution

The answer should have 3 significant figures, because 72.4 has the least number of significant digits(3), and cos58 has many.
I multiplied 72.4 with cos58 using a calculator, and it gave me the result 38.366154... I rounded up to 3 figures, and found (38.4m). When i entered the question on wolfram alpha, it gave me the real result, 38.37m.

 

[brainpushups]

"The least number of significant figures in any number of the problem determines the number of significant figures in the answer."
cos58=0.529919264233...

Is the angle a measurement? If so, then it has two significant figures as written and the answer should have two significant figures. Although the uncertainty is not stated explicitly in either case let us assume that there is an uncertainty of ±1 in the most precise digit in both measurements (that is, 72.4±0.1m and 58±1 degree). The highest possible value given this range of uncertainty is about 39.4 meters. The lowest possible value is about 37.2 meters. The best value is, of course, about 38.4 meters. Notice the variation in the 1s place. The most appropriate way to write the answer would therefore be: 38 meters. Which means (roughly speaking) 38±1m.

Edit: for clarification

 

[trsnd]

Thank you for your answer, but can't we substitue cos58 for 0.529919264233... which is an irrational number, so it has infinite significant numbers, and has no uncertainity? In that case, shouldn't the answer be 38.4?
I don't know, maybe the textbook I'm reading gave the answer wrong, but it's also 38.37 on wolfram alpha.

 

[brainpushups]

It depends. Is 58 degrees a measurement or not? If it is a measurement then it is subject to uncertainty and, as written, has two significant figures. If, on the other hand, you assume that 58 degrees is an exact number then it has 'an infinite number of significant figures' as you say. In that case then your single measurement (the measure of meters) has three significant figures and therefore so must the answer.