Python program to find the sum of sine series

Let us consider that we have a value x and we need to calculate the sum of the sine(x) series. The sine series is represented mathematically as:

sine(x) = x - x³/3! + x?/5! - x?/7! + x?/9! - ...

To solve this series-based problem, we will take the degree as input, convert it to radians, and then iterate through the terms to calculate the sum using the mathematical formula.

Approach

• Take input of the number of terms and degree value.

• Convert degree to radians using the formula: radians = degrees × ?/180

• Iterate over the terms and calculate the sum using power and factorial operations.

• Print the calculated sum.

Example

Here's a complete program to calculate the sine series sum ?

import math

# Input: number of terms and degree value
n = 10
degree = 30

# Convert degree to radians
x = degree * math.pi / 180
print(f"Converting {degree}° to radians: {x:.6f}")

# Initialize variables
sine_sum = 0
sign = 1

# Calculate sine series sum
for i in range(n):
    term = (x ** (2*i + 1)) / math.factorial(2*i + 1)
    sine_sum += sign * term
    sign *= -1  # Alternate signs
    print(f"Term {i+1}: {sign * -1} * {x:.4f}^{2*i+1} / {2*i+1}! = {sign * -1 * term:.8f}")

print(f"\nSum of sine series with {n} terms: {sine_sum:.10f}")
print(f"Built-in math.sin({degree}°): {math.sin(x):.10f}")
print(f"Difference: {abs(sine_sum - math.sin(x)):.2e}")

Converting 30° to radians: 0.523599
Term 1: 1 * 0.5236^1 / 1! = 0.52359878
Term 2: -1 * 0.5236^3 / 3! = -0.02393758
Term 3: 1 * 0.5236^5 / 5! = 0.00033046
Term 4: -1 * 0.5236^7 / 7! = -0.00000206
Term 5: 1 * 0.5236^9 / 9! = 0.00000001
Term 6: -1 * 0.5236^11 / 11! = -0.00000000
Term 7: 1 * 0.5236^13 / 13! = 0.00000000
Term 8: -1 * 0.5236^15 / 15! = -0.00000000
Term 9: 1 * 0.5236^17 / 17! = 0.00000000
Term 10: -1 * 0.5236^19 / 19! = -0.00000000

Sum of sine series with 10 terms: 0.4999999963
Built-in math.sin(30°): 0.5000000000
Difference: 3.70e-09

Simplified Version

Here's a cleaner implementation without detailed output ?

import math

def sine_series(degree, terms=10):
    # Convert degree to radians
    x = math.radians(degree)
    
    sine_sum = 0
    sign = 1
    
    # Calculate sine series
    for i in range(terms):
        term = (x ** (2*i + 1)) / math.factorial(2*i + 1)
        sine_sum += sign * term
        sign *= -1
    
    return sine_sum

# Test the function
degree = 45
calculated_sine = sine_series(degree, 15)
actual_sine = math.sin(math.radians(degree))

print(f"Sine of {degree}° using series: {calculated_sine:.10f}")
print(f"Sine of {degree}° using math.sin(): {actual_sine:.10f}")
print(f"Error: {abs(calculated_sine - actual_sine):.2e}")

Sine of 45° using series: 0.7071067812
Sine of 45° using math.sin(): 0.7071067812
Error: 1.11e-16

How It Works

The sine series converges rapidly for small angles. Each term in the series is calculated using:

• Power calculation: x^(2i+1) for odd powers (1, 3, 5, 7, ...)

• Factorial calculation: (2i+1)! for denominators

• Alternating signs: +, -, +, -, ... pattern

Conclusion

The sine series provides an accurate approximation of the sine function, especially for angles close to zero. More terms in the series yield higher precision, with the error decreasing exponentially for small input values.