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- sin 50 Find all tangent lines at the pole of the graph of. r = -

- sin 50 Find all tangent lines at the pole of the graph of. r = -

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Hello, can you please answer the following question. Please show all your work. Thank you.

### Problem Statement **Goal:** Find all tangent lines at the pole of the graph for the polar equation \( r = -\sin 5\theta \). --- **Solution Steps:** 1. **Understanding the Equation:** - The equation \( r = -\sin 5\theta \) is in polar form. It represents a limaçon. 2. **Analyzing the Points at the Pole:** - The pole is at \( r = 0 \). Therefore, set \( r = -\sin 5\theta = 0 \). - This implies \( \sin 5\theta = 0 \). - Thus, \( 5\theta = n\pi \) for integer \( n \). 3. **Determining \( \theta \):** - Solving for \( \theta \) gives \( \theta = \frac{n\pi}{5} \). 4. **Finding Tangent Lines:** - Each value of \( \theta \) corresponds to a line through the pole: \( \theta = \frac{n\pi}{5} \). 5. **Conclusion:** - The tangent lines at the pole are at angles \( \theta = \frac{n\pi}{5} \), where \( n \) is an integer. This approach helps identify the tangent lines of a polar curve at the origin, which are essential in analyzing the symmetry and features of the curve.
Transcribed Image Text:### Problem Statement **Goal:** Find all tangent lines at the pole of the graph for the polar equation \( r = -\sin 5\theta \). --- **Solution Steps:** 1. **Understanding the Equation:** - The equation \( r = -\sin 5\theta \) is in polar form. It represents a limaçon. 2. **Analyzing the Points at the Pole:** - The pole is at \( r = 0 \). Therefore, set \( r = -\sin 5\theta = 0 \). - This implies \( \sin 5\theta = 0 \). - Thus, \( 5\theta = n\pi \) for integer \( n \). 3. **Determining \( \theta \):** - Solving for \( \theta \) gives \( \theta = \frac{n\pi}{5} \). 4. **Finding Tangent Lines:** - Each value of \( \theta \) corresponds to a line through the pole: \( \theta = \frac{n\pi}{5} \). 5. **Conclusion:** - The tangent lines at the pole are at angles \( \theta = \frac{n\pi}{5} \), where \( n \) is an integer. This approach helps identify the tangent lines of a polar curve at the origin, which are essential in analyzing the symmetry and features of the curve.
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