]> Exercise 2.12

## Exercise 2.12

Set up a spreadsheet to compute the $sin ⁡ x$ for any inputted $x$ . How many terms in the $sin ⁡ x$ power series expansion do you need to evaluate $sin ⁡ .5$ to 8 decimal places?

Solution:

The power series expansion of $sin ⁡ x$ consists of the odd power terms in the expansion of $exp ⁡ x$ , with alternating signs,

$sin ⁡ x = x − x 3 3 ! + x 5 5 ! − ⋯$

We can set up a spreadsheet to compute it in many ways, and here is one.

We will do almost the same thing as done for $exp ⁡ x$ in column A we will put $j$ ; and $j$ will start at 1 and go up by 2 in each successive row. We will put $x$ in the first row of column B and $− x 2$ in successive rows, and multiply previous C entry by entry in B and divide by entry in A and by entry in A, minus 1.

We will get the following formulae in the spreadsheet

 $x$ 0.5 $sin ⁡ x$ 1 =B1 =A2*B2 =D1+C2 =A2+2 =-B2*B2 =C2*B3/A3/(A3-1) =D2+C3 =A3+2 =B3 =C3*B4/A4/(A4-1) =D3+C4 =A4+2 =B4 =C4*B5/A5/(A5-1) =D4+C5 =A5+2 =B5 =C5*B6/A6/(A6-1) =D5+C6 =A6+2 =B6 =C6*B7/A7/(A7-1) =D6+C7 =A7+2 =B7 =C7*B8/A8/(A8-1) =D7+C8 =A8+2 =B8 =C8*B9/A9/(A9-1) =D8+C9 =A9+2 =B9 =C9*B10/A10/(A10-1) =D9+C10 =A10+2 =B10 =C10*B11/A11/(A11-1) =D10+C11 =A11+2 =B11 =C11*B12/A12/(A12-1) =D11+C12 =A12+2 =B12 =C12*B13/A13/(A13-1) =D12+C13 =A13+2 =B13 =C13*B14/A14/(A14-1) =D13+C14 =A14+2 =B14 =C14*B15/A15/(A15-1) =D14+C15 =A15+2 =B15 =C15*B16/A16/(A16-1) =D15+C16 =A16+2 =B16 =C16*B17/A17/(A17-1) =D16+C17 =A17+2 =B17 =C17*B18/A18/(A18-1) =D17+C18 =A18+2 =B18 =C18*B19/A19/(A19-1) =D18+C19 =A19+2 =B19 =C19*B20/A20/(A20-1) =D19+C20 =A20+2 =B20 =C20*B21/A21/(A21-1) =D20+C21 =A21+2 =B21 =C21*B22/A22/(A22-1) =D21+C22

The numerical results will be

 $x$ 0.5 $sin ⁡ x$ 1 0.5 0.5 0.5 3 -0.25 -0.0208333 0.479166667 5 -0.25 0.0002604 0.479427083 7 -0.25 -1.55E-06 0.479425533 9 -0.25 5.382E-09 0.479425539 11 -0.25 -1.223E-11 0.479425539 13 -0.25 1.96E-14 0.479425539 15 -0.25 -2.334E-17 0.479425539 17 -0.25 2.145E-20 0.479425539 19 -0.25 -1.568E-23 0.479425539

You can see that the first 4 terms by themselves give $sin ⁡ .5$ correctly to 8 decimal places.