{"id":224,"date":"2020-12-10T22:53:49","date_gmt":"2020-12-10T22:53:49","guid":{"rendered":"https:\/\/mathdynamics.net\/blog\/?p=224"},"modified":"2021-10-28T19:14:05","modified_gmt":"2021-10-28T19:14:05","slug":"exploring-the-maclaurin-series-expansion-sinx","status":"publish","type":"post","link":"https:\/\/mathdynamics.net\/blog\/index.php\/2020\/12\/10\/exploring-the-maclaurin-series-expansion-sinx\/","title":{"rendered":"Exploring the Maclaurin Series Expansion sin(x)"},"content":{"rendered":"
\n

sin(x) = \\(x – x^3\/3! + x^5\/5! – x^7\/7! + x^9\/9! – \\)…<\/p>\n

The sin(x) is plotted in black.<\/p>\n

Lets break down the amount of work we need to do depending on size of the angle.<\/p>\n

2 Terms are needed for angles 0 through .9 radians<\/p>\n

3 Terms are needed for angles >.9 and <1.57<\/p>\n

4 Terms are needed for angles >1.57 and <2.5 radians<\/p>\n

5 Terms are needed for angles >2.5 and <3.3 radians<\/p>\n

6 Terms are needed for angles >3.3 and <4 radians<\/p>\n

7 Terms are needed for angles >4 and <4.5 radians<\/p>\n

8 Terms are needed for angles >4.5 and <5.5 radians<\/p>\n

9 Terms are needed for angles >5.5 and <=2*pi radians<\/p>\n

This can be A LOT of WORK with pencil and paper.<\/p>\n

Fortunately, the sin of any (A)NGLE > 2*pi will be identical to the sin of an (a)ngle less than or equal to 2*pi.<\/p>\n

So for example, given an (A)ngle of 31.72 radians its corresponding (a)ngle will be .304<\/p>\n

n = int(A\/(2*pi))
\na = A-2*n*pi<\/p>\n

Given A = 31.72 then n = 5
\nhence a = 31.72-(2*5*pi)= .304<\/p>\n

GREAT!! we only need 2 Terms x – x^3\/fact(3)<\/p>\n

Click to explore this Interactive Pallet<\/a>
\n\"Math<\/a><\/p>\n<\/div>\n

\n

This interactive Math Dynamics Pallet demonstrates how the Maclaurin Expansion Series can be used to calculate the sin of any angle<\/p><\/div>\n