Transform (0, y) from rectangular into polar form (here y is any negative number).
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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![**Transform (0, y) from rectangular into polar form (here y is any negative number).**
To convert the point (0, y) from rectangular coordinates (also known as Cartesian coordinates) to polar coordinates, we use the formulas for conversion:
1. **Radius (r):**
\[
r = \sqrt{x^2 + y^2}
\]
Given that \(x = 0\), the formula simplifies to:
\[
r = \sqrt{0 + y^2} = |y|
\]
Since \(y\) is negative, \(r\) will be \(-y\).
2. **Angle (\(\theta\)):**
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
Since \(x = 0\), the tangent function becomes undefined directly. However, for points on the y-axis:
- If \(y\) is negative, \(\theta = \frac{3\pi}{2}\) or \(270^\circ\).
Thus, the polar form of the point (0, y) where \(y\) is negative is \((-y, \frac{3\pi}{2})\) or \((-y, 270^\circ)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83b97d9f-eac5-4c59-98f4-44d1d3519004%2F9189d301-90e9-4f87-bd03-c89f68d57e36%2Fb2sq5hi.png&w=3840&q=75)
Transcribed Image Text:**Transform (0, y) from rectangular into polar form (here y is any negative number).**
To convert the point (0, y) from rectangular coordinates (also known as Cartesian coordinates) to polar coordinates, we use the formulas for conversion:
1. **Radius (r):**
\[
r = \sqrt{x^2 + y^2}
\]
Given that \(x = 0\), the formula simplifies to:
\[
r = \sqrt{0 + y^2} = |y|
\]
Since \(y\) is negative, \(r\) will be \(-y\).
2. **Angle (\(\theta\)):**
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
Since \(x = 0\), the tangent function becomes undefined directly. However, for points on the y-axis:
- If \(y\) is negative, \(\theta = \frac{3\pi}{2}\) or \(270^\circ\).
Thus, the polar form of the point (0, y) where \(y\) is negative is \((-y, \frac{3\pi}{2})\) or \((-y, 270^\circ)\).
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We have convert rectangular coordinates (0 , y) into polar coordinates:
Where y is negative number-
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