The small angle approximation is often made to simplify derivations and calculat

Question

The small angle approximation is often made to simplify derivations and calculations. [See the discussion in the lab manual.] For the following angles ?, compare the true value of the sine of the angle to the to the small angle approximation (using only the first term in the series expansion of the sine) by determining the fractional error (let the fractional error be positive). NOTE: If you explicitly use &pi (pi); to convert from degrees to radians, use an accurate value for &pi (pi);.

DIGRESSION: The fractional error is the difference between the approximate or measured value and the "known or true" value divided by the known or true value. The percent error is the fractional error expressed in percent (percent error = 100%

Explanation / Answer

Using taylor expansion theorem... as you know.. sin(x)= x- x^3/3! + x^5/5! -x^7/7!.... but, as you have said in the question to use only the 1st term, thus.. we are going to use, x (but have to convert it into radians.. ) so.. for a)?=1.26(degree) = 1.26 x 0.0174532925 radians = 0.021991 radian... real value of sin(1.26)=0.02198937 difference= 0.00000163 average= 0.021990185 fractional difference=0.00000741 (approx) b)?=5.66(degree) =0.098767(radians) real value of sin(5.66)=0.098625 difference=0.000142 average=0.098696 fractional difference=0.001438 c)?=16.86(degree) =0.294207(radian) real value of sin(16.86)=0.290034 difference=0.004173 average=0.2921205 fractional difference=0.01428 d)?=43.21(degree) =0.7540145(radian) real value of sin(43.21)=0.6846743 difference=0.0693402 average=0.7193444 fractional value=0.9639 as you can see.. the fractional value increases with the increase in angle... so for more accurate calculation.. u have to go beyond the 1st term of the taylors expansion series.. like till x^7 sin(x)=x- x^3/3! + x^5/5! -x^7/7! in this case, the error , thus you would get would be x^9/9! ( or less than that... but certainly not more than that)... {NOTE:- I compared the values with the 1st term of the taylor series with the real value of sine.. }

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