Find the first three terms of the Taylor series expansion for sin(x) about the point
𝑎
=
𝜋
and use these three terms to estimate the value of sin(3.17)

Enter your answer to two decimal places, rounding as appropriate.

Hint: some of your terms may evaluate to zero

Question

Find the first three terms of the Taylor series expansion for sin(x) about the point 
𝑎
=
𝜋
 and use these three terms to estimate the value of sin(3.17)

Enter your answer to two decimal places, rounding as appropriate.

Hint: some of your terms may evaluate to zero

BlackTom AI · Accepted Answer

Since sin(π) = 0 and cos(π) = −1, the Taylor polynomial about a = π simplifies greatly.

1) Relevant concepts
- Taylor series of f(x) about a: f(a) + f'(a)(x−a) + f''(a)/2!(x−a)^2 + f'''(a)/3!(x−a)^3 + ...
- For f(x)=sin x, derivatives cycle: f = sin, f' = cos, f'' = −sin, f''' = −cos, etc.
- When evaluating at a = π many derivative values are 0 or ±1, so several terms vanish.

2) Step-by-step computation of the first three terms
- f(π) = sin π = 0
- f'(π) = cos π = −1
- f''(π) = −sin π = 0
Thus the first three terms (constant, linear, quadratic) are:
f(π) + f'(π)(x−π) + f''(π)/2 (x−π)^2 = 0 + (−1)(x−π) + 0 = −(x−π).

3) Numerical substitution for x = 3.17
- Compute x − π = 3.17 − 3.141592653589793 = 0.028407346410207
- Approximate sin(3.17) ≈ −(x−π) = −0.028407346410207 ≈ −0.028407

4) Error estimate / higher-order term check
- The next nonzero derivative is f'''(π) = −cos π = 1, so the cubic term is (1/6)(x−π)^3.
- Compute (x−π)^3 ≈ (0.028407)^3 ≈ 2.29×10^−5; divide by 6 gives ≈ 3.82×10^−6, negligible compared with two-decimal rounding.
- Including that would give −0.028404, which does not change the rounded value.

5) Why alternatives would be incorrect
- Using only the constant term (0) ignores the dominant linear behavior and gives a worse estimate.
- A quadratic term here is zero (f''(π)=0), so any quadratic correction is not available; including higher terms changes the estimate by micro amounts only.
- Any sign error (e.g., giving +0.03) would be wrong because cos(π)=−1 makes the linear term negative.

6) Conclusion
Rounded to two decimal places, sin(3.17) ≈ −0.03, which matches the provided answer −0.03.

Find the first three terms of the Taylor series expansion for sin(x) about the point 𝑎 = 𝜋 and use these three terms to estimate the value of sin(3.17) Enter your answer to two decimal places, rounding as appropriate. Hint: some of your terms may evaluate to zeroShort answer

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