Taylor series of sin x up to degree 7
The Taylor polynomial of degree 7 for sin x about 0 (the Maclaurin polynomial) is x - x³/6 + x⁵/120 - x⁷/5040. The full series converges for every x.
How to work it out
- 1
The Taylor series of f about 0 uses the derivatives of f at 0:
- 2
For f(x) = sin x the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 7:
- 4
Euler's formula: e^{iθ} = cos θ + i sin θ. The cos series collects the even powers of the series and the sin series the odd powers.
- 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
\[ \sin\left(x\right) \approx x - \frac{1}{6}x^{3} + \frac{1}{120}x^{5} - \frac{1}{5040}x^{7} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -1,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 1,\quad f^{(6)}(0) = 0 \]
\[ e^{i\theta} = \cos\theta + i\sin\theta \]Degree 7 polynomial against the real sin x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.099833417 | 0.099833417 | 2.75e-15 |
| 0.5 | 0.47942553 | 0.47942554 | 5.37e-9 |
| 0.9 | 0.78332585 | 0.78332691 | 1.06e-6 |
| 2 | 0.90793651 | 0.90929743 | 1.36e-3 |
Questions
What is the Taylor series of sin x?
x - x³/6 + x⁵/120 - x⁷/5040 + …
What is the degree 7 Taylor polynomial of sin x?
x - x³/6 + x⁵/120 - x⁷/5040
What is the radius of convergence of the series of sin x?
∞
What is the Maclaurin series?
A Taylor series centred at 0.