# Significant figures: Law of cosines

• MHB
• SweatingBear
In summary, the problem at hand is to calculate the length of |AB| in a right triangle with sides of length 735m and 420m, and an intermediate angle of 50°. After calculating |AB|, the approximate value is 565.48m. However, there is uncertainty about how many significant digits to include in the answer, as the given data of 420 and 50 are ambiguous in terms of significant digits. The general practice is to have as many digits as the given data with the least amount of significant digits. In this case, it could be interpreted as 420 having either two or three significant digits, and 50 having either one or two significant digits. Ultimately, it is suggested to err on
SweatingBear
We have a right triangle with sides of length 735 and 420 m where the intermediate angle of aforementioned sides is 50°. The task is simple: Calculate |AB|. Here is a picture:

View attachment 1024

Upon calculating |AB|, we arrive at approximately 565.48 m. My problem is how many significant figures one ought to have in the answer. General practice is to have as many digits as the given data with least amount of significant digits.

But in this case, how many digits can we view the data 420 and 50 respectively to have? Integral values with zeros preceding the decimal point can at times be quite ambiguous. It would not make any sense to say "9 000 000" has seven significant digits since it is given without context.

So, 420 either has two significant digits or three; 50 either has one significant digit or two. 565.48 with three significant digits is 565, two significant digits 570 and one significant digit 600. My intuition tells me to answer 565, but I am really not sure which cases of significant digits to confidently rule out.

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sweatingbear said:
We have a right triangle with sides of length 735 and 420 m where the intermediate angle of aforementioned sides is 50°. The task is simple: Calculate |AB|. Here is a picture:

View attachment 1024

Upon calculating |AB|, we arrive at approximately 565.48 m. My problem is how many significant figures one ought to have in the answer. General practice is to have as many digits as the given data with least amount of significant digits.

But in this case, how many digits can we view the data 420 and 50 respectively to have? Integral values with zeros preceding the decimal point can at times be quite ambiguous. It would not make any sense to say "9 000 000" has seven significant digits since it is given without context.

So, 420 either has two significant digits or three; 50 either has one significant digit or two. 565.48 with three significant digits is 565, two significant digits 570 and one significant digit 600. My intuition tells me to answer 565, but I am really not sure which cases of significant digits to confidently rule out.

I suggest to err on the side of caution.
The least significant is the $50^\circ$, which without any extra information we should interpret as $50.0 \pm 0.5^\circ$.

As a result the answer would be approximately $565 \pm 5\text{ m}$.
To err on the side of caution, I would indeed write this down as $565\text{ m}$, which leaves the actual precision still somewhat ambiguous, but at least not as less than the original measurements.

Note that in actual lab work (where precision is important), the standard error of each measurement is recorded.
The way the errors propagate is analyzed and the final results are reported with a specification of the expected error.
If we assume errors of $\pm 0.5\text{ m}$ respectively $\pm 0.5^\circ$, analysis shows that the final error would be $\pm 3.7\text{ m}$.

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Thanks for the reply.

Frankly the problem is merely a simple trigonometry problem and no such where the considering of measurement uncertainty, specifying error bounds or similar courses of actions are necessary to heed (although debatable, but I digress).

So you would take it that 50 is the data with least significant digits? All right I am with you on that, but the significant figures are ambiguous. One significant digit is, in my eyes, an exaggerated (and rather erroneous) approximation of the length. It seems two significant digits is the way to go, but my gut still would have wanted me to answer 565 as opposed to 570.

sweatingbear said:
Thanks for the reply.

Frankly the problem is merely a simple trigonometry problem and no such where the considering of measurement uncertainty, specifying error bounds or similar courses of actions are necessary to heed (although debatable, but I digress).

So you would take it that 50 is the data with least significant digits? All right I am with you on that, but the significant figures are ambiguous. One significant digit is, in my eyes, an exaggerated (and rather erroneous) approximation of the length. It seems two significant digits is the way to go, but my gut still would have wanted me to answer 565 as opposed to 570.

For merely a simple trigonometry problem, it's not really important.
So go with your gut.

Either way, it is ambiguous what the precision of either 565 or 570 is.
To properly specify a precision of 2 digits without going into specifying error bounds, you're supposed to write $5.7 \cdot 10^2\text{ m}$.
But yeah, for a simple trigonometry problem that is over the top.

As a scientist, it is important to use the correct number of significant figures in calculations to accurately reflect the precision of the data. In this case, we can determine the number of significant figures by looking at the given data and determining the least precise measurement.

Since the side lengths of the triangle are given to the nearest meter, we can assume that the data has two significant figures. Therefore, the answer should also have two significant figures. In this case, the answer should be rounded to 570 m.

However, it is always important to consider the context of the data and use your scientific judgment to determine the appropriate number of significant figures. In this case, if the data was given with more precision (e.g. 735.2 and 420.3), then the answer should also reflect that level of precision and have three significant figures.

In conclusion, the answer to this problem should be 570 m, rounded to two significant figures based on the precision of the given data.

## What is the significance of significant figures in the Law of Cosines?

The concept of significant figures is used to determine the precision and accuracy of a measurement. In the Law of Cosines, significant figures are important because they indicate the number of digits that should be used in a calculation to reflect the precision of the measurements being used.

## How do you determine the number of significant figures in a measurement?

The general rule for determining significant figures is that all non-zero digits are significant. Zeros between non-zero digits are also significant. However, leading and trailing zeros may or may not be significant. In the Law of Cosines, only the digits that represent the measurements being used are considered significant.

## What is the purpose of rounding when using significant figures in the Law of Cosines?

In the Law of Cosines, rounding is used to ensure that the final answer reflects the precision of the measurements being used. When performing calculations with significant figures, the final answer should be rounded to the same number of significant figures as the measurement with the least number of significant figures.

## How do significant figures affect the final answer in a Law of Cosines calculation?

The number of significant figures used in a calculation can affect the final answer. If the measurements used have different numbers of significant figures, the final answer should be rounded to the same number of significant figures as the measurement with the least number of significant figures. This ensures that the final answer accurately reflects the precision of the measurements being used.

## Are there any exceptions to the rules of significant figures in the Law of Cosines?

Yes, there are some exceptions to the rules of significant figures in the Law of Cosines. For example, when multiplying or dividing, the final answer should be rounded to the same number of significant figures as the measurement with the least number of significant figures. However, in addition and subtraction, the final answer should be rounded to the same number of decimal places as the measurement with the least number of decimal places.

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