Pseudo Code for Sine Function Evaluation using Infinite Series

Overview

This pseudo code aims to compute the sine of an angle ( x ) using its Taylor series expansion. The algorithm will evaluate terms in the series up to a specified order ( n ), printing the cumulative sine value as each term is added, and calculating the percent relative error compared to the true value of sine.

Parameters

• x: The angle in radians for which sine needs to be computed.
• n: The order up to which the series will be computed.

Constants

• TRUE_SIN: The true sine value computed using a built-in sine function for error calculation.

Steps

1. Input value for x and n
2. Initialize variables
   • sin_x: Initialize to 0 (to store cumulative sine value)
   • factorial: Initialize to 1 (to compute factorial for each term)
   • power: Initialize to x (to compute power of x for each term)
   • sign: Initialize to 1 (to alternate signs in the series)
3. Compute TRUE_SIN
4. Loop from i = 0 to n - 1
   1. Compute term
      • Term = sign * (power / factorial)
   2. Update sin_x
      • sin_x = sin_x + term
   3. Print current term, cumulative sin_x, and TRUE_SIN
   4. Compute percent relative error
      • percent_error = 100 * abs((TRUE_SIN - sin_x) / TRUE_SIN)
   5. Print percent relative error
   6. Update power, sign, and factorial for the next term
      • power = power * x * x (x^2 for the next term)
      • sign = -sign (alternate sign for the series)
      • factorial = factorial * ((2 * i + 2) * (2 * i + 3)) (update factorial for (2i + 2)! = (2i + 2) * (2i + 1)!)

Pseudo Code

FUNCTION compute_sine(x, n)
    TRUE_SIN = sin(x)  // Compute true sine value

    sin_x = 0          // Initialize cumulative sine value
    factorial = 1      // Initialize factorial for the first term
    power = x          // Initialize power of x for the first term
    sign = 1           // Initialize sign for the series

    FOR i FROM 0 TO n - 1 DO
        term = sign * (power / factorial)  // Calculate current term
        sin_x = sin_x + term                // Update cumulative sine

        PRINT "Term", i, ":", term           // Print the current term
        PRINT "Cumulative sin_x:", sin_x    // Print cumulative sine value
        percent_error = 100 * abs((TRUE_SIN - sin_x) / TRUE_SIN)  // Calculate percent error
        PRINT "Percent Relative Error:", percent_error  // Print percent error

        power = power * x * x                // Update power for the next term
        sign = -sign                          // Alternate sign
        factorial = factorial * ((2 * i + 2) * (2 * i + 3))  // Update factorial
    END FOR
END FUNCTION

Conclusion

This pseudo code provides a clear and systematic way to compute the sine of an angle using its Taylor series up to a specified order. The algorithm efficiently retains clarity while outlining each computational step necessary for evaluating both the sine value and its corresponding percent relative error.

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