Series Expansion for sin(x) and cos(x)

Maclaurin Series Expansion

The Maclaurin series is a representation of a function as an infinite sum calculated from the derivatives of the function at a single point.
The general form of a Maclaurin series for a function f(x) is f(x) = f(0) + f’(0)x + f’‘(0)x²/2! + f’’‘(0)x³/3! + …
The term f^n(0)/n! refers to the nth derivative of the function evaluated at x = 0, divided by n factorial.

Expansion for sin(x)

The Maclaurin series expansion for sin(x) is sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + …
The pattern for the expansion of sin(x) consists of alternating plus and minus signs, and the powers on x are odd numbers. The denominator of each term is the factorial of the corresponding power of x.

Expansion for cos(x)

The Maclaurin series expansion for cos(x) is cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + …
The pattern for the expansion of cos(x) also involves alternating plus and minus signs, but the powers on x are even numbers. As with sin(x), the denominator of each term is the factorial of the corresponding power of x.

Application and Significance

These series expansions are useful in approximating function values, especially when the argument of the function is very small (close to zero).
By using a given number of terms from the series, accurate approximations of sin(x) and cos(x) can be obtained.
These expansions play a significant role in a number of mathematics and physics problems, pertaining to wave motion and harmonic analysis, amongst others.

Remember: Practise these expansions to gain proficiency and develop your ability to recognise when they are useful. With a strong understanding of these concepts, tackling complex problems involving sine and cosine functions will become much easier.