Deriving the Expression for sin 2x Using the Taylor Series for sin x

Introduction to Taylor Series for sin x

The Taylor series is a fundamental concept in calculus used to express functions as a sum of infinite series. For the function sin x, the Taylor series is given by:

sin x x - (x^3 / 3!) (x^5 / 5!) - (x^7 / 7!) ...

Each term in this series is centered around x 0, also known as the Maclaurin series. The series converges to the value of sin x for all real numbers x.

Using the Taylor Series for sin x to Derive sin 2x

To derive sin 2x using the Taylor series for sin x, we substitute 2x for x in the series. Let's see how this works:

sin 2x can be expressed using the Taylor series for sin x as follows:

sin 2x (2x) - [(2x)^3 / 3!] [(2x)^5 / 5!] - [(2x)^7 / 7!] ...

Expanding each term, we get:

sin 2x 2x - (8x^3 / 6) (32x^5 / 120) - (128x^7 / 5040) ...

Further simplifying the coefficients:

sin 2x 2x - (4x^3 / 3) (4x^5 / 15) - (4x^7 / 315) ...

This series is indeed the Taylor series for sin 2x, providing an alternative method to the double angle theorem which states sin 2x 2sin x cos x.

Using the Double Angle Theorem

Another way to derive sin 2x is through the double angle theorem:

sin 2x 2sin x cos x

This trigonometric identity can be used when you need a more intuitive understanding of the relationship between sin x and sin 2x. While it is less complex, it may not be as straightforward for deep analytical purposes.

Combining Both Approaches

For some applications, you might find it useful to combine both methods. By multiplying the series for 2sin x and cos x and collecting terms with the same power of x, you would obtain the Taylor series for sin 2x. This approach not only reinforces the understanding of the Taylor series but also highlights the relationship with trigonometric functions.

Conclusion

Both the Taylor series and the double angle theorem provide powerful tools for understanding and manipulating trigonometric functions. The Taylor series method, while more complex, offers a deeper insight into the function's behavior, whereas the double angle theorem provides a more straightforward trigonometric identity.

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