Fixed Point Iteration Calculator for Sin 3 Degrees Ti84

What is Fixed Point Iteration?

Fixed point iteration is a method for finding successively better approximations to the roots (or fixed points) of a function. The basic idea is to start with an initial guess and repeatedly apply a function to get closer to the solution.

The method works by defining a sequence where each term is obtained by applying a function g(x) to the previous term. The sequence converges to a fixed point if the function g(x) is a contraction mapping.

For trigonometric functions, we can use fixed point iteration to approximate values that might not be directly available on a calculator.

Calculating sin(3 degrees)

Calculating sin(3 degrees) directly on a TI-84 is straightforward, but understanding the fixed point iteration method provides insight into how such calculations are performed.

First, we need to convert degrees to radians since most calculators use radians for trigonometric functions. The conversion factor is π/180.

Once we have the angle in radians, we can use the sin function on the TI-84 calculator. However, if we wanted to implement this using fixed point iteration, we would need to define an appropriate function g(x).

TI-84 Instructions

To calculate sin(3 degrees) on your TI-84 calculator:

1. Press the MODE key and ensure that the calculator is in degree mode (DEG).
2. Press the 2ND key and then the SIN key to access the sin function.
3. Enter the angle: 3.
4. Press the ENTER key to get the result.

If you need to perform fixed point iteration for sin(3 degrees), you would typically use a programming approach on the TI-84. Here's a simplified example of how you might set it up:

1. Press the PRGM key and select NEW to create a new program.
2. Name the program (e.g., SINITER).
3. Enter the following code:

:0→A
:0→B
:While abs(A-B)>1E-6
:sin(3)→A
:sin(A)→B
:End
:Disp A

This program initializes two variables A and B, then iterates until the difference between them is very small, indicating convergence. The final value of A will be approximately sin(3 degrees).

Example Calculation

Let's walk through a simple example of fixed point iteration to approximate sin(3 degrees).

We'll use the function g(x) = sin(x) and start with an initial guess of x₀ = 0.05.

After a few iterations, we can see that the values are converging toward approximately 0.05, which is close to the actual value of sin(3 degrees).