Problem 32 - Calculus

Question

Estimate how close \(\theta\) should be to 0 to make \((\sin \theta) / \theta\) stay within 0.001 of 1.

Step-by-Step Solution

Answer

\(|\theta| \approx 0.0775\) ensures \((\sin \theta)/\theta\) stays within 0.001 of 1.

Step 1: Understand the Objective

We're asked to determine how close \(\theta\) should be to 0 for \(\frac{\sin \theta}{\theta}\) to be within 0.001 of 1. This means we want \(1 - 0.001 \leq \frac{\sin \theta}{\theta} \leq 1 + 0.001\).

Step 2: Use Taylor Series

The Taylor series expansion of \(\sin \theta\) around 0 is \(\theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \cdots\). Using this, \(\frac{\sin\theta}{\theta} = 1 - \frac{\theta^2}{6} + \frac{\theta^4}{120} - \cdots\).

Step 3: Make an Approximation

We approximate \(\frac{\sin \theta}{\theta}\) using the first few terms: \(\frac{\sin \theta}{\theta} \approx 1 - \frac{\theta^2}{6}\). This approximation is valid for small \(\theta\).

Step 4: Set Up the Inequality

Using the approximation, set the inequality: \(1 - \frac{\theta^2}{6} = 1 \pm 0.001\). This gives us two cases: \(-0.001 = -\frac{\theta^2}{6}\) and \(0.001 = -\frac{\theta^2}{6}\).

Step 5: Solve for \(\theta\)

Solve \(-0.001 = -\frac{\theta^2}{6}\) which gives \(\theta^2 = 0.006\). Taking the square root, \(|\theta| = \sqrt{0.006} \approx 0.0775\).

Step 6: Form the Conclusion

Since \(\theta\) must be close to 0 for our approximation to hold, the range \(-0.0775 \leq \theta \leq 0.0775\) ensures the expression remains within 0.001 of 1.

Key Concepts

sin(theta)/theta approximation

Approximating \( \frac{\sin \theta}{\theta} \) is crucial when dealing with small angles. This function notoriously approaches 1 as \( \theta \) approaches 0. We use the Taylor series to expand \( \sin \theta \). The series is:

• \( \sin \theta = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \cdots \)

From this series, we derive the following approximation for our expression:

• \( \frac{\sin \theta}{\theta} \approx 1 - \frac{\theta^2}{6} \)

This approximation becomes very accurate when \( \theta \) is small.
It is used in various fields, simplifying complex trigonometric calculations.Understanding and applying this approximation helps solve problems efficiently.

small angle approximation

The small angle approximation is a core concept in trigonometry. It is often used when angles are close to zero. In small angle approximations:

• \( \sin \theta \approx \theta \)
• \( \cos \theta \approx 1 \)

For these approximations to hold, \( \theta \) must be measured in radians.
As angles approach zero, their trigonometric functions can be approximately equal to simpler expressions. This is useful in physics and engineering, where small angles frequently occur.Using small angle approximation:

• Helps simplify calculations.
• Reduces the complexity of equations.

It allows us to manage problems that, without it, might seem too complicated.

inequality solving

Inequality solving is essential for determining the range of values for different expressions. In this exercise, we solve the inequality derived from the approximation:

• Find \( 1 - 0.001 \leq \frac{\sin \theta}{\theta} \leq 1 + 0.001 \)

This inequality is simplified using the approximation of \( \frac{\sin \theta}{\theta} \). So, we test \( 1 - \frac{\theta^2}{6} \) against these bounds.
This results in two separate inequalities:

• \(-0.001 = -\frac{\theta^2}{6}\)
• \(0.001 = -\frac{\theta^2}{6}\)

Solving each inequality leads us to find \( \theta \). These calculations show the range where \( \theta \) must lie, ensuring accuracy within the desired range.Understanding and appropriately applying inequalities is crucial in mathematics and various applications.