Find The Polar Coordinates Of A Point With Cartesian Coordinates (x,y)=(53/2,5/2).Select The Correct

Find The Polar Coordinates Of A Point With Cartesian Coordinates (x,y)=(53/2,5/2).Select The Correct is a common problem in coordinate geometry that helps in understanding the relationship between Cartesian and polar coordinate systems. Converting from Cartesian coordinates (x, y) to polar coordinates (r, θ) involves determining two key values: the radius \( r \), which measures the distance from the origin to the point, and the angle \( \theta \), which indicates the direction of the point relative to the positive x-axis. This process is fundamental in various fields such as physics, engineering, and mathematics, especially when dealing with systems where angles and distances are more meaningful than x and y coordinates.

In this article, we will explore the step-by-step method of converting Cartesian coordinates to polar coordinates, examine the significance of each component, and provide guidance on selecting the correct polar coordinates for the given point. Whether you're a student preparing for exams or a professional working with coordinate transformations, understanding this conversion process is essential.

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Understanding Cartesian and Polar Coordinates

What Are Cartesian Coordinates?

Cartesian coordinates are a two-dimensional coordinate system where any point in the plane is specified by an ordered pair \((x, y)\). The x-coordinate indicates the point's horizontal position, while the y-coordinate indicates its vertical position. For example, the point \((53/2, 5/2)\) is located at \(26.5\) units along the x-axis and \(2.5\) units along the y-axis.

What Are Polar Coordinates?

Polar coordinates describe a point's position in the plane based on its distance from the origin and its angle relative to the positive x-axis. The polar coordinate system is represented as \((r, \theta)\), where:

• \( r \geq 0 \) is the radius or the distance from the origin.
• \( \theta \) is the angle measured in radians or degrees, typically in the range \([0, 2\pi)\) or \([0^\circ, 360^\circ)\).

Converting between these two systems allows for easier analysis in problems involving circles, rotations, and angular measurements.

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Converting Cartesian Coordinates to Polar Coordinates

Step 1: Calculate the Radius \( r \)

The radius \( r \) is the straight-line distance from the origin to the point \((x, y)\). It is computed using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \]

Applying this to our point \((x, y) = (53/2, 5/2)\):
\[
x = \frac{53}{2} = 26.5, \quad y = \frac{5}{2} = 2.5
\]
\[
r = \sqrt{(26.5)^2 + (2.5)^2} = \sqrt{702.25 + 6.25} = \sqrt{708.5}
\]
\[
r \approx 26.62
\]

Step 2: Calculate the Angle \( \theta \)

The angle \( \theta \) is the arctangent of the ratio of \( y \) to \( x \): \[ \theta = \arctan\left(\frac{y}{x}\right) \]

Using our values:
\[
\theta = \arctan\left(\frac{2.5}{26.5}\right) \approx \arctan(0.09434)
\]
Calculating this:
\[
\theta \approx 0.0941 \text{ radians}
\]

Since the point is in the first quadrant (both x and y are positive), this angle directly corresponds to the position relative to the positive x-axis.

To express \( \theta \) in degrees:
\[
\theta \approx 0.0941 \times \frac{180^\circ}{\pi} \approx 5.39^\circ
\]

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Choosing the Correct Polar Coordinates

Considering the Range of \( \theta \)

In polar coordinates, the angle \( \theta \) is often represented within a specific range:

• In radians: \([0, 2\pi)\)
• In degrees: \([0^\circ, 360^\circ)\)

Since our calculated \( \theta \) is approximately \( 0.0941 \) radians or \( 5.39^\circ \), it falls within these ranges easily.

Multiple Representations of Polar Coordinates

Note that a point in the plane can have multiple polar coordinate representations because adding \( 2\pi \) radians (or \( 360^\circ \)) to the angle yields the same point:

• \( (r, \theta) \)
• \( (r, \theta + 2\pi) \)
• \( (-r, \theta + \pi) \) (if considering negative radius)

However, the most standard form is to keep \( r \geq 0 \) and \( \theta \) within \([0, 2\pi)\).

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Final Polar Coordinates of the Point

Based on the calculations:

• \( r \approx 26.62 \)


• \( \theta \approx 0.0941 \) radians or \( 5.39^\circ \)

Thus, the polar coordinate representation of the point \((53/2, 5/2)\) is approximately:
\[
\boxed{
\left( 26.62, \; 0.0941 \text{ radians} \right)
}
\]
or in degrees:
\[
\boxed{
\left( 26.62, \; 5.39^\circ \right)
}
\]

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Summary and Key Takeaways

• The conversion from Cartesian to polar coordinates involves calculating the distance \( r \) using the Pythagorean theorem.
• The angle \( \theta \) is obtained via the arctangent of \( y/x \), considering the quadrant of the point.
• The resulting polar coordinates provide a different perspective, emphasizing the point's distance and direction from the origin.
• Always verify your angle's range and consider multiple representations if needed.

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Practice Problems

1. Convert the Cartesian point \((7, 24)\) to polar coordinates.
2. Find the polar coordinates of \((-3, -3\sqrt{3})\).
3. Given the polar coordinates \((10, \pi/3)\), find the equivalent Cartesian coordinates.

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Conclusion

Mastering the conversion from Cartesian to polar coordinates enhances your understanding of coordinate systems and prepares you for more advanced topics in mathematics and physics. Remember to carefully compute both the magnitude and the angle, considering the quadrant and range conventions. With practice, this process becomes intuitive and essential for analyzing points and systems in two-dimensional space.

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References:

• Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.


• Anton, H., Bivens, I., & Davis, S. (2012). Calculus: Early Transcendentals. Wiley.