Problem 11 - Algebra 2 and Trigonometry

Question

In later courses, you will learn that the sine function can be written as the sum of an infinite sequence. In particular, for \(x\) in radians, the sine function can be approximated as the finite series: $$\sin x \approx x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}$$ a. Graph \(Y_{1}=\sin x\) and \(Y_{2}=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}\) 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 sine approximation is \(-\frac{x^{7}}{7 !}\) . Repeat part a using \(Y_{1}\) and \(Y_{3}=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}\) . 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 sine function values below. Which function gives a better approximation? Is this what you expected? Explain. (1) \(\sin \frac{\pi}{6}\) \(\qquad\) (2) \(\sin \frac{\pi}{4}\) \(\qquad\) \((3) \sin \pi\)

Step-by-Step Solution

Answer

For small values of \(x\), \(Y_2\) is accurate around \(-1\) to \(1\), and \(Y_3\) is good around \(-2\) to \(2\). \(Y_3\) generally provides a better approximation, as expected.

Step 1: Graph Y1 and Y2

Use a graphing calculator to plot the functions \( Y_1 = \sin x \) and \( Y_2 = x - \frac{x^3}{3!} + \frac{x^5}{5!} \). Observe the graphs over a reasonable interval, such as \(-\pi\) to \(\pi\), to see where \(Y_2\) closely follows \(Y_1\). Noting where the approximation diverges from \(Y_1\) provides an understanding of the approximation's limits.

Step 2: Analyze Y2 Approximation

Analyze the graphs of \(Y_1\) and \(Y_2\) to determine where \(Y_2\) closely approximates \(Y_1\). Generally, for small values of \(x\) (around \(-1\) to \(1\)), \(Y_2\) is a good approximation. As \(x\) moves further from zero, especially beyond \([-1, 1]\), the approximation diverges more.

Step 3: Graph Y1 and Y3

Graph the functions \( Y_1 = \sin x \) and \( Y_3 = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \) using the calculator. Again, observe over \(-\pi\) to \(\pi\) to observe how \(Y_3\) compares to \(Y_1\). Determine the interval where \(Y_3\) closely matches \(Y_1\).

Step 4: Analyze Y3 Approximation

Examine the graphs of \(Y_1\) and \(Y_3\). Generally, \(Y_3\) is a good approximation for \(Y_1\) over a slightly larger interval compared to \(Y_2\), perhaps from \(-2\) to \(2\), because the added term increases accuracy.

Step 5: Calculate Sine Values and Compare Approximations

Using \(Y_2\) and \(Y_3\), approximate the values of \(\sin \frac{\pi}{6}, \sin \frac{\pi}{4}, \) and \(\sin \pi\). Calculate:1. \(Y_2\left(\frac{\pi}{6}\right)\) and \(Y_3\left(\frac{\pi}{6}\right)\).2. \(Y_2\left(\frac{\pi}{4}\right)\) and \(Y_3\left(\frac{\pi}{4}\right)\).3. \(Y_2(\pi)\) and \(Y_3(\pi)\).Compare these to actual values: \(\sin \frac{\pi}{6} = \frac{1}{2}, \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \sin \pi = 0\). Notice that \(Y_3\) tends to provide a more accurate approximation, as it includes more terms.

Key Concepts

Understanding the Sine Function

The sine function, denoted as \( \sin x \), is a fundamental element in trigonometry. It is periodic, meaning it repeats its values in regular intervals or cycles. In a geometric sense, the sine of an angle in a right triangle represents the ratio of the length of the opposite side to the hypotenuse. In calculus and beyond, the sine function plays a crucial role especially in the context of waveforms, oscillations, and modeling cyclical phenomena.

The sine function is defined for all real numbers and is typically measured in radians. It has a range of \([-1, 1]\) and a period of \(2\pi\). This function forms the backbone of various applications, from simple trigonometric calculations to complex Fourier series in signal processing. Understanding its foundational properties is key to grasping more complex mathematical concepts.

Infinite Sequence in Taylor Series

A Taylor Series is a way of representing a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For the sine function, the Taylor Series allows us to express \( \sin x \) as an infinite sequence of its derivatives at zero.

This series captures the behavior of \( \sin x \) around the origin and is given by the formula: \[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots \] The '!' symbol denotes factorial, which means multiplying the number by all positive integers less than itself (e.g., \(5! = 5 \times 4 \times 3 \times 2 \times 1\)).

By selecting a finite number of terms from this infinite sequence, we create an approximation of the sine function that retains accuracy within a specific interval. The more terms used from the series, the closer the approximation is to the actual sine function, especially near the origin.

Utilizing a Graphing Calculator

Graphing calculators are invaluable tools in visualizing mathematical functions and their approximations. When graphing the sine function and its Taylor Series approximations (\( Y_1, Y_2, Y_3 \)), we can see how closely the approximations follow the actual sine function.

By plotting \( Y_1 = \sin x \) and its approximations (e.g., \( Y_2 = x - \frac{x^3}{3!} + \frac{x^5}{5!} \)), students can identify where the approximation accurately matches the sine curve. Typically, this involves selecting an appropriate interval such as \(-\pi\) to \(\pi\) to assess where these functions hug the sine curve closely and where they diverge.

• For small values of \(x\), the approximation is generally very close to \(\sin x\).
• As \(x\) moves away from zero, observing how well additional terms refine the curve provides insight into the series' behavior.

This visual method reinforces understanding of how infinite series approximate curved functions effectively.

Enhancing Approximation Accuracy

When working with approximations, accuracy is crucial. In terms of the Taylor Series, adding more terms increases the approximation’s accuracy. Each term captures more of the sine function's characteristic curve and behavior.

For example, when we compare approximations:

• \( Y_2 = x - \frac{x^3}{3!} + \frac{x^5}{5!} \) captures the curve well near the origin but loses precision as \(x\) increases.
• \( Y_3 = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \) offers better precision over a wider interval due to the additional term.

A key component of using Taylor Series effectively is determining the required degree of accuracy for your specific application, balancing computational efficiency with the level of precision needed.

Understanding these trade-offs and estimating the potential error can guide students in making informed choices about the number of terms to use for practical applications.