Approximate the specified function value as indicated and...

Approximate the specified function value as indicated and check your work by comparing your answer to the function value produced directly by your calculating utility.
Approximate $\sin 85^{\circ}$ to four decimal-place accuracy using an appropriate Taylor series.

Key Concepts

Taylor Series

A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. It is used to approximate functions by polynomials, with accuracy increasing as more terms are included. This method serves as a foundation for many numerical approximation techniques in calculus and analysis.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series where the expansion point is zero. It expresses functions as an infinite sum of powers of the variable and is particularly useful when approximating functions near zero. Many common functions, like sine and cosine, have well-known Maclaurin series representations.

Degree to Radian Conversion

Since Taylor series expansions are generally developed using the variable measured in radians, converting degrees to radians is essential when approximating trigonometric functions for angles given in degrees. This conversion is achieved by multiplying the degree measure by ?/180.

Trigonometric Function Approximation

Approximating trigonometric functions like sine using their Taylor or Maclaurin series involves replacing the function with its polynomial series. This method provides a practical way to compute function values to a desired degree of accuracy without relying entirely on calculators.

Error Estimation in Series Approximations

Error estimation involves analyzing how closely a partial sum of a Taylor series approximates the actual function value. Techniques such as the Lagrange remainder provide bounds on the error, ensuring that the approximation meets the required precision, as in attaining four-decimal-place accuracy.

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