Final answer:
To find the 3rd iteration value of x using the Regula-Falsi method for the function f(x) = x - csc(x), compute function values at intervals, use the Regula-Falsi formula to iteratively find a better approximation, and then round off the 3rd iteration value to 5 decimal places while ensuring to carry sufficient precision through intermediate steps.
Step-by-step explanation:
To use the Regula-Falsi method to approximate the solution to f(x) = x - csc(x) on the interval [1,2] and find the 3rd iteration value of x, we follow these steps:The Regula-Falsi method uses the formula: x_{n+1} = x_{n} - f(x_{n}) * (x_{n} - x_{n-1}) / (f(x_{n}) - f(x_{n-1})), where x_n and x_{n-1} are the current and previous approximations respectively, and f(x_{n}) and f(x_{n-1}) are the function values at these points.
Calculations should be carried to several decimal places beyond the desired accuracy to minimize rounding errors, then round-off the 3rd iteration value to 5 decimal places as required.
Important Note
When calculating intermediate values, it is important to maintain precision and round off the final answer to the required significant figures, which ensures that the final approximation is as accurate as possible given the iterative nature of the method.