Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Power series for trig functions

Calculate the coefficients of the Taylor series for the functions cos(θ) and tan(θ) through the order θ4. Sketch graphs of each of the functions together with the approximations provided by the truncation of the power series you have obtained.

For what values of θ is an approximation of O(θ) good for these functions? an approximation of O(θ3)?

Solution

To get the Taylor series for a function we need its derivatives. These are:

so the Taylor series for the two functions are:

The accuracy of the one-term and two-term Taylor series approximations are shown in the figures below.

so the Taylor series for the two functions are:

The constant approximation for cos is within 5% out to about 0.3 rad (almost 20o!) and within 10% out to about 0.5 rad.
The two term series for cos is within 1% to about 0.65 rad and within 10% out to about 1.0 rad.
The three term series for cos is within 1% out to almost 1.2 radians.

The linear term for tan is within 1% to about 0.65 rad and within 10% out to about 0.55 rad.
The two term series for tan is within 1% to about 0.53 rad and within 10% out to about 0.9 rad.

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