Série de Maclaurin sen(x)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
Maclaurin sen
About this calculator
The Maclaurin Series for sin(x) calculator approximates the sine of an angle (in radians) using a power series centered at zero. The formula uses odd terms: sin(x) ≈ x − x³/3! + x⁵/5! − x⁷/7! + ... . The more terms included, the more accurate the approximation. This tool is useful for understanding how Taylor series work in practice, especially for calculations where a scientific calculator is not available.
How it works: the user inputs the value of x (in radians) and chooses the number of series terms (from 1 to 10). The calculator sums the terms according to the formula, alternating signs and computing factorials. The result is displayed along with the exact value of sin(x) (using JavaScript's sine function) for comparison. This allows visualizing the series convergence for different x values.
When to use: this calculator is ideal for calculus students learning Taylor and Maclaurin series, or for professionals needing a quick sine approximation without using trigonometric functions. It is especially useful for small x values, where the series converges quickly. For example, for x near zero, few terms already provide good accuracy.
Cautions: the Maclaurin series for sin(x) converges for all real x, but accuracy depends on the number of terms. For large x (e.g., |x| > π), many terms are needed for a good approximation. Also, the calculator uses radians, not degrees. Make sure to convert degrees to radians (multiply by π/180) before using. Truncation error increases with x and decreases with more terms.
Frequently asked questions
Do I need to convert degrees to radians before using?
Yes, the calculator expects the value in radians. To convert, multiply degrees by π/180 (approximately 0.0174533).
How many terms should I use for a good approximation?
It depends on x. For |x| < 1, 3 to 5 terms already provide good accuracy. For larger x, use more terms (up to 10) to reduce error.
Does the series work for negative angles?
Yes, the Maclaurin series for sine is an odd function, so it works for negative x. The result will be the negative sine of the corresponding positive angle.
Why does the result differ from the exact value?
The difference is the truncation error, as the infinite series is approximated by a finite number of terms. The more terms, the smaller the error.
What is the maximum x value I can use?
There is no theoretical limit, but for |x| > 10, the series converges slowly and may require many terms for accuracy. It is recommended to use x between -π and π for best results.