Problem 64 - Thomas Calculus

Question

The approximation sin \(x \approx x\) It is often useful to know that, when \(x\) is measured in radians, \(\sin x \approx x\) for numerically small values of \(x .\) In Section \(3.11,\) we will see why the approximation holds. The approximation error is less than 1 in 5000 if \(|x|<0.1\) . \begin{equation} \begin{array}{l}{\text { a. With your grapher in radian mode, graph } y=\sin x \text { and }} \\ {y=x \text { together in a viewing window about the origin. What }} \\ {\text { do you see happening as } x \text { nears the origin? }} \\\ {\text { b. With your grapher in degree mode, graph } y=\sin x \text { and }} \\ {y=x \text { together about the origin again. How is the picture }} \\\ {\text { different from the one obtained with radian mode? }}\end{array} \end{equation}

Step-by-Step Solution

Answer

In radian mode, \( \sin x \) and \( x \) nearly overlap, unlike in degree mode.

Step 1: Graphing in Radian Mode

First, ensure that your grapher is set to radian mode. Plot the functions y = \sin x y = x on the same graph. Set the x-axis to a range that includes the origin and where \( |x| < 0.1 \), such as \(-0.5\) to \(0.5\). Observe that as \(x\) approaches the origin, the two curves closely overlap, indicating that \( \sin x \approx x \).

Step 2: Graphing in Degree Mode

Switch your grapher to degree mode and plot the same functions y = \sin x y = x. Use the same x-axis range around the origin. You will notice a significant discrepancy between the two curves near the origin. This is because the approximation \( \sin x \approx x \) holds under radian measure, not degree measure.

Step 3: Comparing Radian and Degree Graphs

Compare the graphs from both radian and degree mode. In radian mode, \( \sin x \) and \( x \) overlap closely near the origin, validating the approximation. In degree mode, there is a visible gap between \( \sin x \) and \( x \), showing that the approximation is not valid, as the scaling of the angle measurement affects the approximation.

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry, representing a ratio of sides in a right triangle. It's defined as the ratio of the length of the side opposite the angle to the hypotenuse. In the context of a circle, sin(x) corresponds to the y-coordinate of a point on the unit circle. This seemingly simple function is immensely powerful, acting as a building block for various advancements in science and engineering.

Mathematically, the sine function is periodic, meaning it repeats at regular intervals. The period of sin(x) is 2π, which explains why the sine wave rises and falls between -1 and 1. The ability to model waveforms is crucial in fields like physics, notably in sound waves and light waves.

The approximation sin(x) ≈ x is most effective when x is close to zero and measured in radians, where understanding radians—a unit of angular measure—is essential for the next topic.

Radians vs Degrees

Understanding radians versus degrees is crucial in trigonometry. Degrees and radians are both units for measuring angles, with 360 degrees in a circle and 2π radians. The conversion between these two is a fundamental step when calculating approximations or solving angular equations.

Radians provide a more natural measure of angles concerning mathematics and physics due to their definition using the radius and arc length of a circle. Essentially, one radian equals the angle created when the arc length equals the radius of the circle. This intrinsic relationship means that many mathematical formulas become simpler when radians are used instead of degrees.

In our sine function approximation, radians are needed because the approximation sin(x) ≈ x at small values of x is based on the Taylor series expansion, valid in radian measure. This highlights why degree mode does not yield the same precision, leading to greater errors as seen when graphing these functions in degree mode.

Graphical Analysis

Graphically analyzing the functions y = sin(x) and y = x can significantly deepen your understanding of their relationship, especially when considering the approximation sin(x) ≈ x for small x.

In radian mode, when graphing y = sin(x) and y = x together, both functions nearly coincide near the origin. This closeness in their graph is emblematic of the approximation's accuracy for small x values (where |x| < 0.1). This proximity visually communicates how the sine function approaches a linear relationship with x in this region.

Switching from radian to degree mode alters the graph severely. Here, the sine curve deviates more noticeably from the line y = x as it approaches the origin. This discrepancy arises because degree mode doesn't inherently preserve the small-angle approximation, signified by a visible gap between the functions.

Through graphical analysis, it's clear why radians are preferred for precise mathematical approximations and how visual tools can illuminate complex concepts.