Problem 8 - Calculus

Question

(Calculator) Compare \(\cos \theta\) with \(1-\frac{1}{2} \theta^{2}\) for (a) \(\theta=0.1\) (b) \(\theta=0.5\) (c) \(\theta=30^{\circ}\) (d) \(\theta=3^{\circ}\).

Step-by-Step Solution

Answer

The approximation is close for small angles, especially in radians, such as \(\theta=0.1\). For larger angles or degrees, the difference becomes more noticeable.

Step 1: Calculate Cosine Value for θ

Use a calculator to find the value of \( \cos \theta \) for each given angle \( \theta \). Make sure your calculator is set to the correct mode (radians or degrees) based on the angle.

Step 2: Calculate Approximation for 1-θ²/2

Calculate the value of \( 1 - \frac{1}{2} \theta^2 \) for each given \( \theta \). Use radians for \( \theta = 0.1 \) and \( \theta = 0.5 \), and degrees for \( \theta = 30^{\circ} \) and \( \theta = 3^{\circ} \).

Step 3: Compare the Two Values

For each angle \( \theta \), compare the values obtained from \( \cos \theta \) and \( 1 - \frac{1}{2} \theta^2 \). Identify which one is greater or if they are approximately equal.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry. It's used to determine the horizontal distance, or adjacent side, when projecting an angle onto a unit circle. The cosine of an angle, denoted as \( \cos(\theta) \, \), is a periodic function with a range between -1 and 1. This periodic nature makes it ideal for modeling wave-like phenomena. The cosine function value changes depending on whether the angle \( \theta \) is measured in degrees or radians. It's important to remember:

• When using degrees, the cycle from 0 to 360°.
• In radians, the full circle is 0 to \( 2\pi \).

Understanding these differences ensures that you correctly interpret cosine values when performing calculations in various areas of mathematics and physics.

Taylor Series

The Taylor series offers a way to approximate complex functions with polynomials, making calculations more manageable. It does so by summing infinite derivatives of the function evaluated at a point. For the cosine function, the Taylor series expansion around 0 is:

• \( \cos(\theta) \, = \, 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots \)

In our exercise, we use the first two terms \( 1 - \frac{\theta^2}{2} \) to approximate the cosine function. This simplification is quite effective for small values of \( \theta \) as it helps to demonstrate the cosine's behavior without using complex calculations. By examining how close these approximations are to actual values, students can gain a practical understanding of how Taylor series can simplify mathematical tasks effectively.

Degrees vs Radians

When discussing angles, choosing between degrees and radians is crucial for accuracy and consistency. Degrees are more intuitive for everyday use, such as directional navigation or basic geometry, but in mathematics, especially calculus, radians are preferred for their natural ties to the unit circle and their simpler formulations in calculus.The conversion between degrees and radians is straightforward; since \( 360° = 2\pi \), you can convert degrees to radians by multiplying by \( \frac{\pi}{180} \). Conversely, to change radians to degrees, multiply by \( \frac{180}{\pi} \).

• A common misstep is not setting your calculator to the correct mode, leading to incorrect results.
• Always verify the mode setting when switching between degrees and radians during calculations.

Understanding when and why to use degrees versus radians helps simplify mathematical operations and reduces the chance of errors in trigonometric calculations.