Problem 49 - Calculus: Early Transcendentals

Question

Use the Maclaurin series for \( \cos x \) to compute \( \cos 5^o \) correct to five decimal places.

Step-by-Step Solution

Answer

\( \cos 5^o \approx 0.99241 \).

Step 1: Recall the Maclaurin Series for \( \cos x \)

The Maclaurin series for the cosine function \( \cos x \) is given by the infinite sum: \[\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\]This series alternates with increasing powers of \(x\) and factorial denominators.

Step 2: Convert Angle from Degrees to Radians

To use the Maclaurin series, the angle must be in radians. Convert \(5^o\) to radians:\[5^o = \frac{5 \pi}{180} = \frac{\pi}{36}\].This is the angle equivalent in radians that we will use in the series.

Step 3: Substitute \(x = \frac{\pi}{36}\) into the Series

Substitute \(x = \frac{\pi}{36}\) into the Maclaurin series:\[\cos \left(\frac{\pi}{36}\right) = 1 - \frac{(\frac{\pi}{36})^2}{2!} + \frac{(\frac{\pi}{36})^4}{4!} - \frac{(\frac{\pi}{36})^6}{6!} + \cdots\]We will compute several terms until the sum converges to a value correct to five decimal places.

Step 4: Compute the First Few Terms

Calculate each term up to the one contributing less than \(0.00001\):- First term: \(1\).- Second term: \(-\frac{(\frac{\pi}{36})^2}{2!} \approx -0.00760\).- Third term: \(+\frac{(\frac{\pi}{36})^4}{4!} \approx 0.00001\).Continue to approximate until the next term is less than \(0.00001\).

Step 5: Sum the Terms until Desired Accuracy is Achieved

Add the first few terms:- \(1 - 0.00760 + 0.00001 \approx 0.99241\).The effects of further terms are negligible (less than \(0.00001\)), thus the computation is correct to five decimal places.

Key Concepts

Taylor series

A Taylor series provides an approximation of functions, particularly around a specific point. While a Maclaurin series is a special case of a Taylor series, where the center point is zero. The essence of a Taylor series is to represent a function as an infinite sum of terms computed from the function's derivatives at a single point. For a function, say \( f(x) \), its Taylor series around a point \( a \) is represented as:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots\]

• The derivatives provide the values for each term.
• Factorials in the denominators scale down the term contributions as the series progresses.

In practice, we often use a finite number of terms from the series to achieve a desired accuracy in approximation.

trigonometric functions

Trigonometric functions, such as cosine (\( \cos \)), sine (\( \sin \)), and tangent (\( \tan \)), are foundational elements in mathematics, especially in geometry and calculus. These functions relate to angles in right-angled triangles and circles. The cosine function, specifically, represents the ratio of the adjacent side to the hypotenuse in a right triangle.

• Cosine is even: \( \cos(-x) = \cos(x) \).
• It oscillates between -1 and 1.

The Maclaurin series we're employing uses the cosine function to model it using mathematical terms that are easier to work with in certain calculations, like small angle approximations.

angle conversion in radians

Working with trigonometric functions often requires angles to be in radians, especially when applying calculus or utilizing series like the Maclaurin series. Radians provide a natural way of measuring angles using the circle's radius. In this context, 360 degrees is equivalent to \( 2\pi \) radians, and vice versa.To convert degrees to radians, use the relation:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]When computing the Maclaurin series for \( \cos 5^o \), we first convert 5 degrees to radians by multiplying by \( \frac{\pi}{180} \). This is critical for accurate results because series calculations assume angles are in radians.

series convergence

Series convergence is a vital concept when working with infinite series, like Taylor or Maclaurin series. It refers to how closely the partial sums of the series come to the value of the function being represented. For convergence:

• Each subsequent term in the series should become smaller in magnitude.
• The series should approach a finite limit.

In computing \( \cos 5^o \) to five decimal places, convergence ensures that by summing sufficient terms, we capture the true value accurately. Once terms become small enough (like below a certain threshold, e.g., \( 0.00001 \)), further terms have negligible impact on the overall sum, achieving our desired precision.