Problem 54 - University Calculus: Early Transcendentals

Question

Find the limits. $$\lim _{\theta \rightarrow 0} \cos \left(\frac{\pi \theta}{\sin \theta}\right)$$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Understand the Limit Expression

We need to find \( \lim_{\theta \to 0} \cos \left(\frac{\pi \theta}{\sin \theta}\right) \). This involves the limit of the cosine function as \( \theta \) approaches 0. We must simplify the expression or find relevant limits to determine this.

Step 2: Simplify the Inner Fraction

First, simplify \( \frac{\pi \theta}{\sin \theta} \). As \( \theta \to 0 \), the small angle approximation \( \sin \theta \approx \theta \) can be used. Thus, \( \frac{\pi \theta}{\sin \theta} \approx \frac{\pi \theta}{\theta} = \pi \).

Step 3: Substitute Limit in Cosine Function

Now, with the simplification from Step 2, we substitute in the cosine function: \( \cos \left(\frac{\pi \theta}{\sin \theta}\right) \approx \cos(\pi) \).

Step 4: Evaluate the Cosine Function

Since \( \cos(\pi) = -1 \), the limit of the cosine function as \( \theta \to 0 \) becomes \(-1\).

Step 5: Final Conclusion

Thus, the limit as \( \theta \to 0 \) of our original expression is \(-1\). Therefore, \( \lim _{\theta \rightarrow 0} \cos \left(\frac{\pi \theta}{\sin \theta}\right) = -1 \).

Key Concepts

Trigonometric Limits

Trigonometric limits are essential when evaluating expressions involving sine, cosine, and other trigonometric functions as a variable approaches a certain value. These limits evaluate how a function behaves close to a certain input without actually reaching that input. For the cosine function, we often assess its limit as its argument approaches zero or other crucial angles.Recognizing limits of well-known values is helpful:

• \( \lim_{\theta \to 0} \sin \theta = 0 \)
• \( \lim_{\theta \to 0} \cos \theta = 1 \)

In this context, knowing these basic limits helps us understand more complex expressions, such as in the exercise we have, where trigonometric functions converge to known values. Evaluating these limits unfolds their behaviors, making it easier to predict how they interact in compounded mathematical expressions.

Small Angle Approximation

The small angle approximation is a valuable tool in calculus and trigonometry, useful when dealing with trigonometric functions as their angles approach zero. This approximation simplifies calculations, making it easier to find limits.As \( \theta \to 0 \), it is known that:

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

These approximations are derived because the Taylor series for sine and cosine around zero starts with these linear relationships. In our exercise, using \( \sin \theta \approx \theta \) allows us to simplify \( \frac{\pi \theta}{\sin \theta} \approx \frac{\pi \theta}{\theta} = \pi \).Employing such approximations provides a straightforward way to predict behavior and outcomes without detailed calculations and decreases computational complexity in practical scenarios.

Cosine Function

The cosine function, part of the fundamental trigonometric functions, is periodic with a cycle of \( 2\pi \). It represents the adjacent side over the hypotenuse in a right triangle and has certain key values that are critical for understanding its limits:

• \( \cos(0) = 1 \)
• \( \cos(\pi) = -1 \)
• \( \cos(2\pi) = 1 \)

When solving limits involving the cosine function, it's crucial to understand these outputs. In limits like \( \lim_{\theta \to 0} \cos \left( \frac{\pi \theta}{\sin \theta} \right) \), after simplifying the argument to \( \pi \), we use the fact that \( \cos(\pi) = -1 \), to evaluate the expression.Understanding these properties allows for quick and accurate limit calculations within trigonometric contexts. These insights simplify otherwise complex trigonometric expressions, providing a gateway to deeper mathematical analysis.