Fixed Point Iteration Calculator for Sin 3 Degrees

What is Fixed Point Iteration?

Fixed point iteration is a numerical technique used to find solutions to equations of the form x = g(x). The method involves repeatedly applying a function g(x) to an initial guess until the sequence converges to a fixed point.

The basic steps of fixed point iteration are:

1. Choose an initial guess x₀
2. Compute the next approximation using xₙ₊₁ = g(xₙ)
3. Repeat until the difference between consecutive approximations is smaller than a specified tolerance

How to Use This Calculator

To use this calculator for finding the fixed point of sin(x) at 3 degrees:

1. Enter the initial guess (in degrees)
2. Set the tolerance (how close the approximations need to be to stop)
3. Select the maximum number of iterations
4. Click "Calculate" to see the results

Fixed Point Iteration for sin 3 degrees

For the function sin(x) with an initial guess of 3 degrees, we can find the fixed point using fixed point iteration. The fixed point is the value where x = sin(x).

The iteration formula for this case is:

The calculator will apply this formula repeatedly until the difference between consecutive approximations is smaller than the specified tolerance or the maximum number of iterations is reached.

Example Calculation

Let's walk through an example calculation for sin(3°):

1. Initial guess: x₀ = 3° (0.0524 radians)
2. First iteration: x₁ = sin(3°) ≈ 0.0523 radians (3.000°)
3. Second iteration: x₂ = sin(3.000°) ≈ 0.0523 radians (3.000°)

The sequence quickly converges to approximately 3.000° because sin(3°) is very close to 3°.