how many times does $r = 2 + sin(\theta)$ pass through x = 1 in cartesian coordinates?

To determine how many times the polar curve $ r = 2 + \sin(\theta) $ passes through the line $ x = 1 $ in Cartesian coordinates, we need to translate the problem into a coordinate system that allows us to compare both equations.

### Step 1: Convert Polar to Cartesian Coordinates
In polar coordinates, we know the following relationships between $ r $ and $ \theta $, and the Cartesian coordinates $ x $ and $ y $:

\[
x = r \cos(\theta)
\]
\[
y = r \sin(\theta)
\]

Substituting $ r = 2 + \sin(\theta) $ into the equation for $ x $:

\[
x = (2 + \sin(\theta)) \cos(\theta)
\]

We are looking for the values of $ \theta $ where $ x = 1 $, so set the equation equal to 1:

\[
(2 + \sin(\theta)) \cos(\theta) = 1
\]

### Step 2: Solve the Equation for $ \theta $
To solve $ (2 + \sin(\theta)) \cos(\theta) = 1 $, we can attempt to solve it directly or graphically analyze it to determine how many solutions exist within one period of $ \theta $.

Let's proceed graphically or numerically. The equation depends on trigonometric functions, which have periodic behavior, so there might be multiple values of $ \theta $ within the range $ [0, 2\pi] $ for which this equation holds.

### Step 3: Numerical or Graphical Solution (Discussion)
Without performing a detailed numerical analysis here, it's known that polar

How many times does $r = 2 + sin(\theta)$ pass through x = 1 in cartesian coordinates?

Learn how to determine how many times the polar curve r = 2 + sin(θ) passes through x = 1 in Cartesian coordinates. This problem involves polar to Cartesian conversion, trigonometric identities, and solving trigonometric equations graphically or numerically. Suitable for undergraduate math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation