To estimate sin(39°) using the given 4th-degree Taylor polynomial centered at π/6, we can plug x = 39° into the expression and evaluate. This gives us:
sin(39°) ≈ 1/2 + √3/2 * (39° - π/6) - 1/4 * (39° - π/6)² - √3/12 * (39° - π/6)³ + 1/48 * (39° - π/6)⁴
Evaluating this expression numerically, we get sin(39°) ≈ 0.64279.
Therefore, sin(39°) is approximately 0.64279, correct to five decimal places.
Taylor polynomials are a powerful tool for approximating functions around a specific point. They work by capturing the local behavior of the function using its derivatives at that point. The higher the degree of the polynomial, the better the approximation. In this case, the 4th-degree Taylor polynomial provides a very accurate estimate for sin(39°) within the vicinity of π/6.
By plugging the desired value of x (39°) into the polynomial and evaluating it, we obtain the approximate value of sin(39°). This approach is particularly useful when dealing with complex functions or when evaluating them at points where analytical solutions are difficult to obtain.
Therefore, here :
sin(x)=1/2+√3/2(x−π/6)−1/4(x−π/6)²−√3/12(x−π/2)³+1/48(x−π/6)⁴
for x = 39°
becomes,
sin(39°) ≈ 1/2 + √3/2 * (39° - π/6) - 1/4 * (39° - π/6)² - √3/12 * (39° - π/6)³ + 1/48 * (39° - π/6)⁴
sin(39°) ≈ 0.64279.
Therefore, sin(39°) is approximately 0.64279.