Question

Write the Taylor series expansion of cos(x): Use the first three terms to calculate the value of cos(n/4). Use the decimal format with six significant digits (apply rounding at each step). Calculate the truncation error A- B-

Answer 1

Taylor series expansion of xos x is layer series expansion of f(x) = cos x of about x = 0 is f(x) = f(0) + f^1(0) x/1 + f^11(0)/2x^2 + f^111(0)/3x^3 +.... f(x) = cosx: f(0) = cos(0) = 1 f^1(x) = -sin x f^1(0) = -sin 0 = 0 f^11(x) = -cos x f^11(0) = -1 f^111(0) = sin x: f^111(0) = sin 0 = 0 f^(iv) (x) = cos x: f^(v) cos = cos o = 1 So, taylor series expansion of f(x) = cos x is f(x) = 1 to - x^2/2! - x^3/3!(0) + x^4/4!(1)..... = 1 - x^2/2! + x^4/4!.... f(x) = cos x at x = pi/4 co pi/4 = 1 - (pi/4)^2/2 + (pi/4)^4/4!..... almostequalto 1 - pi^2/16 times 2 + pi^4/256 times 24 almostequalto 1 - 0.308425 + 0.015854 almostequalto - 0.707429 now, we know that w pi/4 = 1/squareroot 2 = 0.7071068 almostequalto 0.707107 Error = 0.707429 - 0.707107 = 0.000322