**Question:**Find the equation of the line tangent to the cardioid $ r = 1 + \sin(\theta) $ at $ \theta = 0 $.---**Solution:**1. **Convert to Cartesian Coordinates:** The polar coordinates $ r $ and $ \theta $ can be converted to Cartesian coordinates $ (x, y) $ using the formulas: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \] For the given cardioid $ r = 1 + \sin(\theta) $, we first calculate the coordinates at $ \theta = 0 $. \[ r = 1 + \sin(0) = 1 + 0 = 1 \] \[ x (0) = 1 \cos(0) = 1 \cdot 1 = 1 \] \[ y (0) = 1 \sin(0) = 1 \cdot 0 = 0 \] Therefore, the point of tangency in Cartesian coordinates is $ (1, 0) $.2. **Find the Derivative:** To find the slope of the tangent line, we consider the derivatives of $ x $ and $ y $ with respect to $ \theta $. \[ x = r \cos(\theta) = \left(1 + \sin(\theta)\right) \cos(\theta) \] \[ y = r \sin(\theta) = \left(1 + \sin(\theta)\right) \sin(\theta) \] Differentiate $ x $ and $ y $ with respect to $ \theta $: \[ \frac{dx}{d\theta} = \frac{d}{d\theta} \left( (1 + \sin(\theta)) \cos(\theta) \right) \] Apply the product rule: \[ \frac{dx}{d\theta} = \cos(\theta) \frac{d}{d\theta} (1 + \sin(\theta)) + (1 + \sin(\theta)) \frac{d}{d\theta} \cos(\theta) \] \[

Name: **Question:**Find the equation of the line tangent to the cardioid $
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: **Question:**Find the equation of the line tangent to the cardioid \( r = 1 + \sin(\theta) $ at $ \theta = 0 $.---**Solution:**1. **Convert to Cartesian Coordinates:** The polar coordinates $ r $ and $ \theta $ can be converted to Cartesian

Given that,, Here, differentiate w.r.t. ,we get, We know that, To find: equation of the line…

Answered: Find the equation of the line tangent…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Topic Video

Question

Expert Solution

Step by step

Solved in 3 steps with 14 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Chain Rule$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated