Maclaurin series and the Taylor’s inequality

Question

How do we use Maclaurin series and the Taylor’s inequality to estimate the value of 4 cos 5° to four decimal-places.

Answer 1

The angle is given in degrees. We need to convert it to radians first.

theta = (5 * pi)/180 = 0.0872665 (approximately)

Now use the Maclaurin series with the first three terms for 4 cos(theta).

4 * cos(theta) = 4 * ( 1 - (theta^2)/2! + (theta^4)/4! )

= 4 * (1-(0.0872665^2)/2 + (0.0872665 ^4)/24)

= 4 * 0.99619469544 = 3.98477878178

the approximation of 4 cos(theta) = 3.98477878178

Use Taylor's inequlity and find an estimate for the error term:

|R} <= (theta^6)/720 for using 4 terms of the series

<= 6.1341283 e-10

The tolerace we use is 0.00005 since we want the result to be accurate to 4 decimal places. so, the upper error bound in our case

is less than the 0.00005 and that shows that our approximation of the 4 cos(theta) is accurate to four decimal places.