Problem 44 - Single Variable Calculus

Question

Use the Maclaurin series for \(\cos x\) to calculate \(\cos 3^{\circ}\) accurate to five decimal places.

Step-by-Step Solution

Answer

The short answer is: To calculate \(\cos 3^{\circ}\) using the Maclaurin series to 5 decimal places, first convert the angle to radians: \(x = \frac{\pi}{60}\). Find the smallest \(n\) for which the absolute value of the n-th term is less than \(10^{-5}\): \(|\frac{(-1)^n}{(2n)!} (\frac{\pi}{60})^{2n}| < 10^{-5}\). Then compute the sum using the Maclaurin series for cos(x): \(\cos \frac{\pi}{60} \approx \sum_{n=0}^{N} \frac{(-1)^n}{(2n)!} (\frac{\pi}{60})^{2n}\). Round the result to 5 decimal places to get the final answer.

Step 1: Convert degrees to radians

In order to use the Maclaurin series formula for cos(x), we need the angle in radians. To convert from degrees to radians, we use the formula: radians = degrees * \(\frac{\pi}{180}\) For an angle of 3 degrees: radians = \(3 \cdot \frac{\pi}{180} = \frac{\pi}{60}\) Now we can substitute this angle into the Maclaurin series formula for cos(x).

Step 2: Determine the number of terms needed for the desired accuracy

To determine the number of terms required to get an accuracy of five decimal places, we can find the smallest n for which the absolute value of the n-th term is less than \(10^{-5}\). We can use the following relation for the Maclaurin series for cos(x): \(|\frac{(-1)^n}{(2n)!}x^{2n}| < 10^{-5}\) Replacing x with \(\frac{\pi}{60}\), we get: \(|\frac{(-1)^n}{(2n)!} (\frac{\pi}{60})^{2n}| < 10^{-5}\) We can now test different values of n until we find the smallest n satisfying this inequality.

Step 3: Calculate the Maclaurin series sum

Using the smallest n found in step 2, we can now calculate the sum of the Maclaurin series for cos(x): \(\cos x \approx \sum_{n=0}^{N} \frac{(-1)^n}{(2n)!} x^{2n}\) For x = \(\frac{\pi}{60}\): \(\cos \frac{\pi}{60} \approx \sum_{n=0}^{N} \frac{(-1)^n}{(2n)!} (\frac{\pi}{60})^{2n}\) Calculate the sum and round the result to 5 decimal places to get the final answer.

Key Concepts

Converting Degrees to Radians

When working with trigonometric functions, it's important to note that most mathematical formulas assume angle measurements are in radians, not degrees. To convert degrees to radians, we use a simple ratio: there are \(2\pi\) radians in a full 360-degree circle, so each degree is equivalent to \(\frac{\pi}{180}\) radians. For an angle of \(x\) degrees, the conversion formula is: \[ radians = x * \frac{\pi}{180} \].

For example, to convert 3 degrees to radians, we calculate \(3 \cdot \frac{\pi}{180} = \frac{\pi}{60}\), and now this radian measure can be used in the Maclaurin series for trigonometric functions.

Approximating Functions with Series

A Maclaurin series is a way to approximate complex functions using an infinite sum of polynomial terms whose coefficients are derived from the function's derivatives at zero. Specifically for \(\cos(x)\), the Maclaurin series is an infinite sum of even-powered terms with alternating signs and factorial denominators. The formula for \(\cos(x)\) is given by: \[ \cos x \approx \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} \].

This series is an example of a power series, which allows us to approximate \(\cos(x)\) for small values of \(x\) with high precision, by calculating a finite number of terms. The closer \(x\) is to zero, the fewer terms are needed to accurately approximate the function.

Determining Series Accuracy

Determining the accuracy of a series approximation involves deciding how many terms to include. If your goal is to achieve a certain level of precision, you can stop adding terms once the absolute value of the next term in the series would no longer affect the decimal place up to which you want precision. For example, for a five decimal place accuracy, one must continue to add terms until adding the next term changes the result by less than \(10^{-5}\).

To assess this, calculate the next term in the series and see if it is smaller than the required precision level. In the case of the Maclaurin series for \(\cos(x)\), we would solve the inequality \(\left| \frac{(-1)^n}{(2n)!} x^{2n} \right| < 10^{-5}\) for \(n\), where \(x\) is the value in radians, to determine how many terms \(N\) are needed.

Calculating Series Sum

To calculate the series sum, especially for trigonometric functions like \(\cos(x)\), we utilize the identified Maclaurin series expansion and the computed number of terms \(N\) that meet our accuracy target. The sum is found by systematically adding each term, from \(n = 0\) up to the highest necessary term \(n = N\), substituting in the radian value for \(x\).

Using our earlier conversion and accuracy determination, the sum for \(\cos(\frac{\pi}{60})\) can be expressed as \[ \cos \left(\frac{\pi}{60}\right) \approx \sum_{n=0}^{N} \frac{(-1)^n}{(2n)!} \left(\frac{\pi}{60}\right)^{2n} \]. We calculate each term up to \(n = N\) and sum them together. Then we round off the result to the desired level of accuracy, in this case, to five decimal places.