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Taylor series of sin x up to degree 1

The Taylor polynomial of degree 1 for sin x about 0 (the Maclaurin polynomial) is x. The full series converges for every x.

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sin x and its degree 1 Taylor polynomial-3-2-1123-1-0.50.51sin xdegree 1
sin x and its degree 1 Taylor polynomial. The polynomial follows the curve closely near 0 and drifts away further out; more terms stretch the good region.

How to work it out

  1. 1

    The Taylor series of f about 0 uses the derivatives of f at 0:

    f(x)=∑k=0∞f(k)(0)k! xkf(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k
  2. 2

    For f(x) = sin x the first derivatives at 0 are:

    f(0)(0)=0,f(1)(0)=1f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1
  3. 3

    Divide each by k! and keep the terms up to degree 1:

    sin⁡(x)≈x\sin\left(x\right) \approx x
  4. 4

    Euler's formula: e^{iθ} = cos θ + i sin θ. The cos series collects the even powers of the exe^{x} series and the sin series the odd powers.

    eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta
  5. 5

    Radius of convergence: ∞. Inside it, more terms give a better approximation; outside, the series does not settle.

LaTeX source of the formulas on this page
\[ \sin\left(x\right) \approx x \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1 \]
\[ e^{i\theta} = \cos\theta + i\sin\theta \]

Degree 1 polynomial against the real sin x

xPolynomialExactError
0.10.100000000.0998334171.67e-4
0.50.500000000.479425542.06e-2
0.90.900000000.783326911.17e-1
22.00000000.909297431.09e+0

Questions

What is the Taylor series of sin x?

x - x³/6 + x⁵/120 - x⁷/5040 + …

What is the degree 1 Taylor polynomial of sin x?

x

What is the radius of convergence of the series of sin x?

∞

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of sin x

Degree 1 of other functions

    Taylor series of sin x: degree 1 polynomial and error | WordGPT