How do you use the Remainder Estimate Series and the series for sin(x) to calculate sin(1) within a hundreth?

Answer 1

The simple series for sin(x) is the McLaurin series which is the Taylor expansion around 0:

sin(x) = x - (1/3!)x^3 + (1/5!)x^5 - (1/7!)x^7 + ...

The rule for a sum of a series with alternating signs is that the error is less than the first term neglected.

For x = 1, x^7 = 1 and 1/7! = 1/840 so we can certainly ignore the x^7 term.

For x = 1, x^5 = 1 and 1.5! = 1/120 so we can ignore the x^5 term, but it would be close, so I'd include it.

So that leaves you with three terms: 1 - 1/6 + 1/120

Answer 2

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