Sin Calculation Formula

Taylor Series Expansion for Sine:

\[ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

Taylor Series Approximation:

Built-in sin(x) Value:

Difference:

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1. What is the Taylor Series for Sine?

The Taylor series expansion for sine is an infinite series that approximates the sine function using polynomial terms. It's particularly useful for calculating sine values without direct trigonometric function calls.

2. How Does the Calculator Work?

The calculator uses the Taylor series expansion:

\[ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

Where:

\( x \) — Input angle in radians
\( n \) — Term index (0, 1, 2, ...)
\( ! \) — Factorial operation

Explanation: The series alternates between positive and negative terms with odd exponents and factorials in the denominators. More terms yield better approximations.

3. Importance of Taylor Series

Details: Taylor series are fundamental in numerical analysis and computer implementations of mathematical functions. They allow calculation of transcendental functions using basic arithmetic operations.

4. Using the Calculator

Tips: Enter the angle in radians (for degrees, multiply by π/180). More terms increase accuracy but require more computation. For most practical purposes, 5-10 terms are sufficient.

5. Frequently Asked Questions (FAQ)

Q1: Why use Taylor series instead of built-in sin()?
A: This demonstrates the mathematical foundation. Built-in functions are optimized and more accurate, but understanding the series helps in numerical methods.

Q2: How accurate is the approximation?
A: Accuracy improves with more terms. Near x=0, convergence is fast. For larger x, more terms are needed.

Q3: What's the radius of convergence?
A: The series converges for all real numbers (infinite radius of convergence).

Q4: Can I use this for degrees?
A: First convert degrees to radians (multiply by π/180 ≈ 0.0174533).

Q5: Why does the difference matter?
A: The difference shows the error in the approximation compared to PHP's built-in sin() function.