Problem 10 - Algebra 2 and Trigonometry

Question

In later courses, you will learn that the cosine function can be written as the sum of an infinite sequence. In particular, for \(x\) in radians, the cosine function can be approximated by the finite series: $$ \cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !} $$ a. Graph \(Y_{1}=\cos x\) and \(Y_{2}=1-\frac{x^{2}}{2 !}+\frac{x^{2}}{4 !}\) on the graphing calculator. For what values of \(x\) does \(Y_{2}\) seem to be a good approximation for \(Y_{1} ?\) b. The next term of the cosine approximation is \(-\frac{x^{6}}{6 !}\) Repeat part a using \(Y_{1}\) and \(Y_{3}=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}\) For what values of \(x\) does \(Y_{3}\) seem to be a good approximation for \(Y_{1} ?\) c. Use \(Y_{2}\) and \(Y_{3}\) to find approximations to the cosine function values below. Which function gives a better approximation? Is this what you expected? Explain. \(\begin{array}{llll}{\text { (1) } \cos -\frac{\pi}{6}} & {\text { (2) } \cos -\frac{\pi}{4}} & {\text { (3) } \cos -\pi}\end{array}\)

Step-by-Step Solution

Answer

Y2 closely approximates Y1 for \( x \in (-\frac{\pi}{3}, \frac{\pi}{3}) \), and Y3 extends this range. Y3 generally gives a better approximation than Y2, as expected.

Step 1: Understand the Cosine Approximation Expressions

The given approximations for the cosine function are finite series representations. The function \( Y_1 = \cos x \) is the standard cosine function. The approximation \( Y_2 = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} \) uses terms up to the degrees of 4, and \( Y_3 = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} \) uses terms up to the degrees of 6. These approximations are derived from the Taylor series expansion of \( \cos x \) around 0.

Step 2: Graphing and Comparison for Y2

Using a graphing calculator, graph \( Y_1 = \cos x \) and \( Y_2 = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} \). Observe where the plots of these functions closely overlap. \( Y_2 \) should closely approximate \( Y_1 \) around \( x = 0 \), typically in the range \( -\frac{\pi}{3} \) to \( \frac{\pi}{3} \). Analyze the graph to see this approximation better around small values of \( x \).

Step 3: Graphing and Comparison for Y3

Now, graph \( Y_1 = \cos x \) and \( Y_3 = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} \). Notice the range where these graphs overlap closely. By adding more terms, \( Y_3 \) will be a good approximation over a wider range around \( x = 0 \) than \( Y_2 \), often extending beyond \( -\frac{\pi}{3} \) to \( \frac{\pi}{3} \).

Step 4: Calculate Using Y2 and Y3

Calculate \( Y_2 \) and \( Y_3 \) for specific values: \( x = -\frac{\pi}{6}, -\frac{\pi}{4}, -\pi \). Substitute these x-values into both approximations to find each value. Then compare them with the actual values of \( \cos x \) to determine which approximation is closer for each x-value.

Step 5: Compare Accuracy and Reflect

Compare the approximations calculated in Step 4 for \( Y_2 \) and \( Y_3 \) with the actual \( \cos x \) values. Observe that as the degree of the approximation polynomial, \( Y_3 \) typically yields better accuracy than \( Y_2 \). Reflect on the expected improvement due to additional higher-order terms introduced in \( Y_3 \).

Key Concepts

Cosine Function

The cosine function, \( \cos x \), is a fundamental trigonometric function with applications in various fields including physics, engineering, and computer graphics. Cosine helps measure the horizontal distance as compared to the radius in a unit circle. It's crucial to understand that while the cosine function is periodic with a cycle of \(2\pi\), it is also smooth and continuous.

When graphed, the cosine function produces a wave-like pattern that starts at its maximum of 1 when \(=0\), dips to a minimum at -1, and then returns to 1. In calculus and higher mathematics, the cosine function can be expressed using an infinite series known as Taylor Series, which allows for approximations of the function around specific points often simplifying computations.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. The concept of an infinite series is widely used in mathematics to represent functions, solve equations, and model various phenomena. Many functions that are difficult to compute directly can be expressed and approximated by an infinite series, such as the Taylor Series or Maclaurin Series.

In the context of the cosine function, it can be expanded into an infinite series as follows:
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots \]
This infinite series representation allows mathematicians and scientists to approximate the cosine function with desired precision by using finite sums, which become manageable for computation and analysis.

Graphing Calculator

Using a graphing calculator is a powerful way to visually understand the behavior of functions. Graphing calculators are especially useful for students trying to visualize how approximations compare with the actual function.

In exercises involving series approximations of functions like cosine, students can enter the equation for the actual cosine function and its approximations (e.g., the finite series approximation \( Y_2\) and \( Y_3\)) into their graphing calculator. This visual graphing offers immediate insight into where the approximation closely matches the true function, typically around the expansion point, such as \(x = 0\) in the case of the Taylor Series expansion for cosine.

Approximation

Approximation is an essential concept in mathematics, often utilized when it is impractical or impossible to compute exact values. By using approximations, one can estimate values of complex functions within acceptable margins of error.

In the exercise, approximations of the cosine function are represented by finite series of terms, allowing a simpler way to calculate the value of \( \cos x \) without using its actual trigonometric function. The effectiveness of an approximation depends on how close its values are to the true values of the function over a specific range. Typically, adding more terms to a series increases the accuracy of the approximation.

Finite Series

The term finite series refers to a sum of a limited number of terms from an infinite series. In pursuing mathematical problems, one often stops at a finite series to provide an approximation that balances accuracy with computational feasibility.

In the context of the cosine function, approximating it with a finite series like \( Y_2 = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} \) allows a simpler, yet effective method to evaluate the function over a specific interval close to zero. The addition of terms reduces errors and can significantly improve approximations over larger intervals, as demonstrated in \( Y_3\) which included a third term: \( -\frac{x^6}{6!} \). These finite series provide means to manage calculations that infinite series would render impractical for straightforward use.