How to Approximate arccos(1/4) by Hand?

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SUMMARY

The discussion focuses on approximating arccos(1/4) using the Taylor Series expansion. The series begins with the formula arccos(x) ≈ π/2 - x - x³/6, leading to an approximation of arccos(1/4) as approximately 1.318. The approximation is validated by calculating (1/4)³/6, which equals 0.003, indicating a close estimate. Additionally, the derivative of f(x) = arccos(x) can be expressed as a power series, which can be integrated and evaluated for convergence at x = 1/4.

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how to approximate arccos(1/4) by hand?
 
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Well the Taylor Series for arccos(x) begins as pi/2 - x - x^3/6, since (1/4)^3/6 = .003 this is a pretty good approximation:

arccos(1/4) \approx \pi/2 - x - x^{3}/3 \approx 1.318

knowing that pi/2 is about 1.57 might help too.
 
The derivative of f(x) = arccos(x) can be written as a power series (a binomial series). Find this series, integrate both sides, check for convergence, and plug in x = 1/4.
 

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