Pseudo Code for Evaluating the Sine Function Using an Infinite Series

Overview

The objective is to evaluate the sine function using the Taylor series expansion up to a specified order ( n ). The algorithm will compute and print the values of ( \sin(x) ) as each term is added, and will also compute and display the percent relative error compared to the actual sine value.

Key Concepts

• Sine Function Taylor Series: The sine function can be approximated by the series: [ \sin(x) = \sum_{k=0}^{n} \frac{(-1)^k x^{2k+1}}{(2k+1)!} ]
• Percent Relative Error: This is calculated as: [ \text{percent_relative_error} = \left| \frac{\text{actual_value} - \text{computed_value}}{\text{actual_value}} \right| \times 100 ]

Pseudo Code

FUNCTION computeSineAndError(x, n)
    DECLARE actualValue AS FLOAT
    DECLARE computedValue AS FLOAT
    DECLARE error AS FLOAT
    DECLARE term AS FLOAT
    DECLARE factorial AS INTEGER
    DECLARE powerFactor AS FLOAT
    DECLARE percentRelativeError AS FLOAT
    
    SET actualValue = sin(x)  // Compute the actual sine value
    
    computedValue = 0         // Initialize computed sine value

    // Loop through each term up to n
    FOR k FROM 0 TO n DO
        IF k == 0 THEN
            term = x            // First term is x^1 / 1!
        ELSE
            // Calculate the factorial and power factor
            factorial = FACTORIAL(2 * k + 1)
            powerFactor = POWER(x, 2 * k + 1)
            term = ((-1)^k * powerFactor) / factorial  // Compute the k-th term

        // Update the computed sine value
        computedValue = computedValue + term
        
        // Print the current term and computed sine value
        PRINT "Term: ", term
        PRINT "Computed sin(x): ", computedValue
        
        // Calculate percent relative error
        IF actualValue != 0 THEN
            percentRelativeError = ABS((actualValue - computedValue) / actualValue) * 100
            PRINT "Percent Relative Error: ", percentRelativeError
        ELSE
            PRINT "Actual value is 0, Relative error is undefined"
        END IF
    END FOR
    
END FUNCTION

FUNCTION FACTORIAL(n)
    DECLARE result AS INTEGER
    SET result = 1
    FOR i FROM 1 TO n DO
        result = result * i
    END FOR
    RETURN result
END FUNCTION 

FUNCTION POWER(base, exponent)
    DECLARE result AS FLOAT
    SET result = 1
    FOR i FROM 1 TO exponent DO
        result = result * base
    END FOR
    RETURN result
END FUNCTION

Explanation

• computeSineAndError(x, n): This function computes the sine of ( x ) and prints out each term of the series, along with the cumulative sine value and the percent relative error.
• FACTORIAL(n): Computes the factorial of ( n ) through iterative multiplication.
• POWER(base, exponent): Raises the base to the power of the exponent using iterative multiplication.

Conclusion

This pseudo code outlines a clear and structured approach to calculating the sine function using its Taylor series expansion, while also allowing for the evaluation of the accuracy of the approximation through the percent relative error.