How to Use Taylor's Theorem to Expand sin(π/6 h) in Ascending Powers of h
In this article, we will explore how to use Taylor's Theorem to expand the function sin(π/6 h) in ascending powers of h, up to the term in h4. This technique is fundamental in numerical analysis, calculus, and applied mathematics. We will walk through the process step-by-step, making it accessible for both students and professionals alike.
The Taylor Expansion Formula
The Taylor expansion formula is a powerful tool that allows us to approximate a function using its derivatives at a specific point. For a function f(x) around a point a, the Taylor expansion up to the fourth derivative is given by:
f(a h) ≈ f(a) f'(a)h frac{1}{2!}f''(a)h^2 frac{1}{3!}f'''(a)h^3 frac{1}{4!}f''''(a)h^4 O(h^5)
Applying Taylor's Theorem to sin(π/6 h)
Let's apply this formula to the function f(x) sin(x) at the point a π/6. We will find the first four derivatives of sin(x) and evaluate them at a π/6. Then, we will substitute these values into the Taylor expansion formula.
Derivatives of sin(x)
f(x) sin(x)
f'(x) cos(x)
f''(x) -sin(x)
f'''(x) -cos(x)
f''''(x) sin(x)
Evaluating Derivatives at π/6
f(π/6) sin(π/6) 1/2
f'(π/6) cos(π/6) √3/2
f''(π/6) -sin(π/6) -1/2
f'''(π/6) -cos(π/6) -√3/2
f''''(π/6) sin(π/6) 1/2
Substituting into the Taylor Expansion Formula
Now, let's substitute the evaluated derivatives into the Taylor expansion formula:
sin(π/6 h) ≈ 1/2 (√3/2)h frac{1}{2!}(-1/2)h^2 frac{1}{3!}(-√3/2)h^3 frac{1}{4!}(1/2)h^4 O(h^5)
Simplifying this expression, we get:
sin(π/6 h) ≈ 1/2 (√3/2)h - 1/4h^2 - √3/12h^3 1/48h^4 O(h^5)
Conclusion
In conclusion, Taylor's Theorem provides a practical method to approximate trigonometric functions like sin(x) around a given point. By evaluating the derivatives and substituting them into the expansion formula, we can obtain a precise approximation of the function in ascending powers of h. This technique is invaluable in a wide range of applications, from physics and engineering to computer graphics and machine learning.
Understanding and mastering Taylor's Theorem is essential for anyone pursuing studies or careers in mathematics, science, and technology. The skills and knowledge gained from this process are not only theoretical but can be directly applied to solve real-world problems.