how to convert tangent to degrees without calculator
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If you take a circular arc of radius rr and arc length ll, then the corresponding angle will be l/rl/r radians. In particular, a radian is just 180/π180/π degrees. This would convert an expression like tan(360/2n)tan(360/2n)working with degrees to one like tan(π/n)tan(π/n) working with radians. This is somewhat unfortunate, since you can't very well plug in ππ if that's the value you're trying to calculate. Note that your method is basically equivalent to saying that the derivative of the tangent function at x=0x=0 is equal to one.
My take is that the inverse tangent function provides a reasonable method for approximating ππ by hand. Machin's formula gives the following expansion:
π==16arctan(1/5)−4arctan(1/239)(45−43∗53+45∗55−47∗53+…)+(−4239+43∗2393−45∗2395+47∗2397−…)π=16arctan(1/5)−4arctan(1/239)=(45−43∗53+45∗55−47∗53+…)+(−4239+43∗2393−45∗2395+47∗2397−…)The justification that this works requires Taylor series, but computing approximations this way is pure algebra.
My take is that the inverse tangent function provides a reasonable method for approximating ππ by hand. Machin's formula gives the following expansion:
π==16arctan(1/5)−4arctan(1/239)(45−43∗53+45∗55−47∗53+…)+(−4239+43∗2393−45∗2395+47∗2397−…)π=16arctan(1/5)−4arctan(1/239)=(45−43∗53+45∗55−47∗53+…)+(−4239+43∗2393−45∗2395+47∗2397−…)The justification that this works requires Taylor series, but computing approximations this way is pure algebra.
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There is a Youtube Video for a better explanation
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