October 19th, 2015, 06:08 PM  #1 
Newbie Joined: May 2014 From: New York Posts: 3 Thanks: 0  Error approximation help
The hypotenuse of a right triangle is known to be 20 in exactly, and one of the acute angles is measured to be 30 degrees with a possible error of plusorminus 2 degrees . (a) Use differentials to estimate the errors in the sides opposite and adjacent to the measured angle. (b) Estimate the percentage errors in the sides. I know on (b) the percentage error of the adjacent side is 2% and the opposite is 6% but I have no idea how to get (a) 
October 19th, 2015, 07:04 PM  #2 
Member Joined: Apr 2015 From: USA Posts: 46 Thanks: 32 
The side opposite from the angle is described by $\displaystyle\sin\theta=\frac{opp}{20}$. Differentiate to get $\displaystyle\cos\theta\,d\theta=\frac{d\,opp}{20 }$ $\displaystyle\theta=\frac{\pi}{6}\qquad \text{Angles have to be in radians.}$ And we're given the error in the angle as $\pm2$, so: $\displaystyle d\theta=2\frac{\pi\text{ radians}}{180\text{ degrees}}=\frac{\pi}{90}$ Now solve for $d\,opp$. You should get 0.605. Ideally, the $opp$ should be length 10. So % error is $\displaystyle\frac{0.605}{10}\cdot100\approx6\%$ You should be able to figure out the adjacent side. 

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approximation, error 
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