Problem 42 Hard Difficulty

On page 431 of $ Physics: Calculus, $ 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/ Cole, 2000), in the course of deriving the formula $ T = 2\pi \sqrt {L/g} $ for the period of a pendulum of length $ L, $ the author obtains the equation $ a_T = -g \sin \theta $ for the tangential acceleration of the bob of the pendulum. He then says, "for small angles, the value of $ \theta $ in radians is very nearly the value of $ \sin \theta; $ they differ by less than $ 2\% $ out to about $ 20^o." $
(a) Verify the linear approximation at 0 for the sine function:
$ \sin x \approx x $
(b) Use a graphing device to determine the value of $ x $ for which $ \sin x $ and $ x $ differ by converting from radians to degrees.

Answer

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Video Transcript

okay, part ay. So the first thing we know is that we can use the Formula 000 plus f prime of zeros one times X minus zero, which gives us acts. So Olive Axe is simply equivalent to X copping for my graphing calculator. Now we know we have X minus Sign axe over. Sign X is 0.2 So to figure out where this equals again using a graphing calculator, we get closer minus your 0.344 times 1 80 over pie gives us 19.7 degrees. So, in other words, remember, this is between negative 0.3440 point 344 because it's plus or minus 3440.344