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 $\cos \left(-175^{\circ}\right)$ to four decimal-place accuracy using a Taylor series.

Key Concepts

Approximation Accuracy and Error Analysis

Approximating a function value using a finite number of terms from its Taylor series introduces an error known as the remainder. To achieve a specific accuracy, such as four decimal places, one must consider the magnitude of the error term and decide on the number of terms needed. Error analysis helps in estimating the truncation error, ensuring that the approximation meets the desired precision when compared with the actual function value.

Degree to Radian Conversion

In calculus, trigonometric functions in Taylor series are typically expressed in terms of radians rather than degrees. Therefore, when a function value is given in degrees, it is essential to convert it to radians before using the Taylor series approximation. This conversion ensures that the input to the series matches the standard trigonometric definitions used in mathematical analysis.

Taylor Series Expansion

The Taylor series is an infinite sum that represents a function as an expansion around a specific point. When approximating functions like cosine, the Taylor series allows us to compute the function's value using a polynomial whose terms involve derivatives evaluated at that point. This method is especially useful when evaluating the function directly is complex or when an approximation to a desired degree of accuracy is acceptable.

Cosine Function and Its Polynomial Approximation

The cosine function, which is periodic and even, can be approximated using its Taylor series expansion. For the cosine function, the series is composed of only even-powered terms with alternating signs, making it particularly convenient for approximations near zero. This polynomial approximation gets more accurate as additional terms are included, allowing one to approximate cosine values to any required level of precision.

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