The 3rd degree Taylor polynomial for cos(x) centered at a = π 2 is given by, cos(x) = − x − π 2 + 1 6 x − π 2 3 + R3(x). Using this, estimate cos(88°) correct to five decimal places.

There are quite a few mistakes in writing the the question, but i corrected them below. firstly the value of a that you stated seems incorrect and based on the last part of the question asking for cos(88°) i figured that what you were trying to write down was a=π/2 , since 88° is in the vicinity of a=π/2 or 90° and thats what the taylor series expansions are all about, that is estimating the values of functions about a given point. The second problem was in writing down the taylor series itself. So i used the formula for the taylor series expansion and obtained the correct version about the point a=π/2 that is,

[tex]cos(x) = cos(pi/2) - sin(pi/2) *(x - pi/2) - (cos(pi/2)(x - pi/2)^{2})/(1*2)) + (sin(pi/2)*(x - pi/2)^{3} /(1*2*3))[/tex]

[tex]cos(x) = (x - pi/2) + (x - pi/2)^{3} /6\\cos(88) = ((88/90)*(pi/2) - pi/2) + ((88/90)*(pi/2) - pi/2)^{3} /6\\cos(88) = 0.034906[/tex]

In the last step we converted degrees into radians and then plugged it in.

If you use the calculator to evaluate cos(88°) you would get the same result to 5 decimal places.