Worksheet 3 Solutions

# Worksheet 3 Solutions - Math 121 Trigonometry Worksheet 3...

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Math 121 – Trigonometry Worksheet 3: Taylor Series Problem 1: Taylor series A Taylor series is like a polynomial, but infinitely long. It is used to approximate values of functions that are generally too difficult to evaluate directly. For example, the Taylor series for sine and cosine are given by the following formulas: sin( x ) = x - x 3 3! + x 5 5! - x 7 7! + · · · = X k =0 ( - 1) k x 2 k +1 (2 k + 1)! cos( x ) = 1 - x 2 2! + x 4 4! - x 6 6! + · · · = X k =0 ( - 1) k x 2 k (2 k )! (a) Use a calculator and the first four terms of these series to approximate sin(1) and cos(1). Compare your answers with what your calculator tells you should be the answer. (Actually, this is how your calculator computes these values, but with more terms for more accuracy.) Solution: Using the first for terms of the Taylor series, we get the following: sin(1) 1 - 1 3! + 1 5! - 1 7! 0 . 841468 cos(1) 1 - 1 2! + 1 4! - 1 6! 0 . 540278 If you use some form of electronic computing device to evaluate sin and cosine at 1, you should get something similar. Using Mathematic, I got: Sin[1.] = 0.841471 Cos[1.] = 0.540302 (The decimal point tells Mathematica that I want a decimal approximation.)

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