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October 19th, 2015, 05:08 PM   #1
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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 plus-or-minus 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)
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October 19th, 2015, 06:04 PM   #2
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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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