Percent Error on Tangent Small Angle Approximation

Calculus Level 3

At what angle in radians to the nearest thousandth $\big($ in the range $[0,2\pi)\big)$ does the relative error between $\tan \theta$ and its lowest-order small-angle approximation exceed 5%?

The answer is 0.385.

1 solution

Matt DeCross
Mar 13, 2016

The lowest order small-angle approximation for $\tan x$ is $x$ . Solving the below equation:

$\frac{\tan x - x}{\tan x} = .05$

numerically (e.g. with WolframAlpha) obtains the solution.

But this is wrong - the error is 0.0499; the correct answer is 0.386

Sebastian Allon - 3 months, 2 weeks ago

I see what the issue is - I computed the exact answer, and then rounded it to nearest thousandth. You took increments of .001 in the angle until you exceeded 5%. Strictly speaking I think the way the question is worded implies you should round the exact answer. But if you got 0.386, you can also consider yourself correct :) If I could add more than one correct answer I would.

Matt DeCross - 3 months, 1 week ago

Percent Error on Tangent Small Angle Approximation

The answer is 0.385.

1 solution

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