Convert The Polar Coordinates \left(6 \sqrt{3}, \frac{\pi}{6}\right ] Into Rectangular Form. Express Your Answer In Simplest Radical Form.
Introduction
Polar coordinates are a way of representing points in a two-dimensional plane using a distance from a reference point (the origin) and an angle from a reference direction (the positive x-axis). Rectangular coordinates, on the other hand, are a way of representing points in a two-dimensional plane using the x and y distances from the origin. In this article, we will discuss how to convert polar coordinates to rectangular form.
What are Polar Coordinates?
Polar coordinates are a way of representing points in a two-dimensional plane using a distance from a reference point (the origin) and an angle from a reference direction (the positive x-axis). The distance from the origin is called the radial distance, and the angle from the positive x-axis is called the angular distance. Polar coordinates are often denoted as (r, θ), where r is the radial distance and θ is the angular distance.
What are Rectangular Coordinates?
Rectangular coordinates are a way of representing points in a two-dimensional plane using the x and y distances from the origin. The x-coordinate is the distance from the origin to the point along the x-axis, and the y-coordinate is the distance from the origin to the point along the y-axis. Rectangular coordinates are often denoted as (x, y).
Converting Polar Coordinates to Rectangular Form
To convert polar coordinates to rectangular form, we can use the following formulas:
x = r cos(θ) y = r sin(θ)
where x and y are the rectangular coordinates, r is the radial distance, and θ is the angular distance.
Example: Converting Polar Coordinates to Rectangular Form
Let's consider the polar coordinates (6√3, π/6). We can use the formulas above to convert these coordinates to rectangular form.
x = 6√3 cos(π/6) y = 6√3 sin(π/6)
To evaluate these expressions, we can use the following trigonometric identities:
cos(π/6) = √3/2 sin(π/6) = 1/2
Substituting these values into the expressions above, we get:
x = 6√3 (√3/2) y = 6√3 (1/2)
Simplifying these expressions, we get:
x = 9 y = 3√3
Therefore, the rectangular coordinates of the point (6√3, π/6) are (9, 3√3).
Simplifying the Rectangular Coordinates
In the previous example, we obtained the rectangular coordinates (9, 3√3). However, we can simplify these coordinates further by factoring out the common factor of 3.
x = 9 = 3(3) y = 3√3 = 3√3
Therefore, the simplified rectangular coordinates are (3(3), 3√3).
Conclusion
In this article, we discussed how to convert polar coordinates to rectangular form. We used the formulas x = r cos(θ) and y = r sin(θ) to convert the polar coordinates (6√3, π/6) to rectangular form. We also simplified the rectangular coordinates further by factoring out the common factor of 3. We hope that this article has provided a clear understanding of how to convert polar coordinates to rectangular form.
Additional Resources
For more information on polar coordinates and rectangular coordinates, please see the following resources:
Final Answer
Introduction
In our previous article, we discussed how to convert polar coordinates to rectangular form. In this article, we will provide a Q&A section to help clarify any questions or doubts that readers may have.
Q: What are polar coordinates?
A: Polar coordinates are a way of representing points in a two-dimensional plane using a distance from a reference point (the origin) and an angle from a reference direction (the positive x-axis). The distance from the origin is called the radial distance, and the angle from the positive x-axis is called the angular distance. Polar coordinates are often denoted as (r, θ), where r is the radial distance and θ is the angular distance.
Q: What are rectangular coordinates?
A: Rectangular coordinates are a way of representing points in a two-dimensional plane using the x and y distances from the origin. The x-coordinate is the distance from the origin to the point along the x-axis, and the y-coordinate is the distance from the origin to the point along the y-axis. Rectangular coordinates are often denoted as (x, y).
Q: How do I convert polar coordinates to rectangular form?
A: To convert polar coordinates to rectangular form, you can use the following formulas:
x = r cos(θ) y = r sin(θ)
where x and y are the rectangular coordinates, r is the radial distance, and θ is the angular distance.
Q: What if I have a polar coordinate with a negative radial distance?
A: If you have a polar coordinate with a negative radial distance, you can convert it to a positive radial distance by adding π to the angular distance. For example, if you have the polar coordinate (r, θ) with r < 0, you can convert it to the polar coordinate (|r|, θ + π).
Q: What if I have a polar coordinate with an angle of π/2?
A: If you have a polar coordinate with an angle of π/2, you can convert it to rectangular form using the following formulas:
x = r sin(Ï€/2) y = r cos(Ï€/2)
Since sin(Ï€/2) = 1 and cos(Ï€/2) = 0, you can simplify the formulas to:
x = r y = 0
Q: Can I convert polar coordinates to rectangular form using a calculator?
A: Yes, you can convert polar coordinates to rectangular form using a calculator. Most calculators have a built-in function to convert polar coordinates to rectangular form. You can enter the polar coordinates (r, θ) and the calculator will output the rectangular coordinates (x, y).
Q: What are some common mistakes to avoid when converting polar coordinates to rectangular form?
A: Some common mistakes to avoid when converting polar coordinates to rectangular form include:
- Forgetting to use the correct formulas (x = r cos(θ) and y = r sin(θ))
- Not simplifying the expressions (e.g. not factoring out common factors)
- Not checking for negative values (e.g. not adding π to the angular distance when r < 0)
Conclusion
In this article, we provided a Q&A section to help clarify any questions or doubts that readers may have about converting polar coordinates to rectangular form. We hope that this article has been helpful in providing a clear understanding of this important concept.
Additional Resources
For more information on polar coordinates and rectangular coordinates, please see the following resources:
Final Answer
The final answer is: