Significant Digits for Longer Operations

[jwj11]

Homework Statement

Ok so I am having difficulty doing significant digits for longer operations (i.e. cosine law) where you also have square roots and squares (I know they are simply multiplications)

Homework Equations

c^2 = a^2 + b^2 - 2ab(cosC)

The Attempt at a Solution

Here are some example numbers. Please let me know if the answer is in correct significant digits.

c^2 = 22^2 + 65^2 - 2(22)(65)(cos119degrees) , c greater than or equal to 0
c=78

I think for cosine law what is really confusing me is how the cos119 converts into a very long decimal (weird fraction results)

And what about for sine law?

example

sin 119 degrees / 78.08 = sin theta / 65

theta = 47 degrees

Are the significant digits for both answers correct? Please help!

 

[dynamicsolo]

I'm not entirely sure I understand what you are concerned about, but you are permitted to use "excessive" significant figures in the course of making a calculation. (People do this all the time...) You simply have to report the final result to the correct number of significant figures.

You'll find in math courses, though, that people tend to be a lot less fussy about this. In a sense, this is because the values given in problems (for lengths, angles, and such) are assumed to be exact, and so have an infinite number of significant figures. You are then permitted to given the answer to whatever number of significant figures seems reasonable...

In fact, it is generally a good idea not to cut down the number of significant figures for an intermediate result. For some functions, this can lead to enormous errors in calculations. An example might be, for the law of cosines problem, using the cosine function for an angle of 89.5º, when the lengths of the sides are only given to two significant figures. Rounding the angle to 89º or 90º would lead to a huge misevaluation of the cosine of the actual given angle.

 

[Moetasim]

When dealing with longer operations involving significant digits, it is important to follow the rules of significant figures. In this case, the final answer should have the same number of significant digits as the measurement with the least number of significant digits. For example, in your first equation, the measurement with the least number of significant digits is 22 (2 significant digits). Therefore, the final answer should also have 2 significant digits. In this case, the final answer should be rounded to 78, not 78.08. Similarly, in your second equation, the measurement with the least number of significant digits is 65 (2 significant digits), so the final answer should also have 2 significant digits, resulting in an answer of 47 degrees. It is important to remember to round your final answer to the appropriate number of significant digits to ensure accuracy.