Convert To Rectangular Coordinates. (Give Exact Answers.)Polar: { (6, -\frac 3\pi}{4})$}$Rectangular { ( \square , \square )$ $
Understanding Polar and Rectangular Coordinates
In mathematics, polar coordinates and rectangular coordinates are two different ways to represent the position of a point in a two-dimensional plane. Polar coordinates are represented as (r, θ), where r is the distance from the origin to the point and θ is the angle between the positive x-axis and the line segment connecting the origin to the point. Rectangular coordinates, on the other hand, are represented as (x, y), where x is the horizontal distance from the origin to the point and y is the vertical distance from the origin to the point.
Converting Polar Coordinates to Rectangular Coordinates
To convert polar coordinates to rectangular coordinates, 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 distance from the origin to the point, and θ is the angle between the positive x-axis and the line segment connecting the origin to the point.
Example: Converting Polar Coordinates to Rectangular Coordinates
Let's consider the polar coordinates (6, -3Ï€/4). We can use the formulas above to convert these coordinates to rectangular coordinates.
x = 6 cos(-3Ï€/4) y = 6 sin(-3Ï€/4)
To evaluate these expressions, we need to use the values of cosine and sine for the angle -3Ï€/4.
cos(-3π/4) = -1/√2 sin(-3π/4) = -1/√2
Now we can substitute these values into the formulas above.
x = 6 (-1/√2) y = 6 (-1/√2)
Simplifying these expressions, we get:
x = -6/√2 y = -6/√2
Rationalizing the Denominator
To rationalize the denominator, we can multiply the numerator and denominator by √2.
x = (-6/√2) (√2/√2) y = (-6/√2) (√2/√2)
This simplifies to:
x = -6√2/2 y = -6√2/2
Simplifying the Expressions
We can simplify these expressions further by dividing both the numerator and denominator by 2.
x = -3√2 y = -3√2
Conclusion
In this article, we have learned how to convert polar coordinates to rectangular coordinates using the formulas x = r cos(θ) and y = r sin(θ). We have also seen how to evaluate these expressions using the values of cosine and sine for the angle -3π/4. Finally, we have rationalized the denominator and simplified the expressions to get the final answer.
Key Takeaways
- Polar coordinates are represented as (r, θ), where r is the distance from the origin to the point and θ is the angle between the positive x-axis and the line segment connecting the origin to the point.
- Rectangular coordinates are represented as (x, y), where x is the horizontal distance from the origin to the point and y is the vertical distance from the origin to the point.
- To convert polar coordinates to rectangular coordinates, we can use the formulas x = r cos(θ) and y = r sin(θ).
- We can evaluate these expressions using the values of cosine and sine for the angle θ.
- We can rationalize the denominator and simplify the expressions to get the final answer.
Practice Problems
- Convert the polar coordinates (3, π/4) to rectangular coordinates.
- Convert the polar coordinates (4, -Ï€/3) to rectangular coordinates.
- Convert the polar coordinates (2, 3Ï€/2) to rectangular coordinates.
Answer Key
- (3/√2, 3/√2)
- (-4√3/2, -4/2)
- (0, -2)
References
- "Polar Coordinates" by Math Open Reference
- "Rectangular Coordinates" by Math Open Reference
- "Converting Polar Coordinates to Rectangular Coordinates" by Wolfram MathWorld
Converting Polar Coordinates to Rectangular Coordinates: Q&A ===========================================================
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about converting polar coordinates to rectangular coordinates.
Q: What are polar coordinates?
A: Polar coordinates are a way of representing the position of a point in a two-dimensional plane using a distance from the origin (r) and an angle from the positive x-axis (θ).
Q: What are rectangular coordinates?
A: Rectangular coordinates are a way of representing the position of a point in a two-dimensional plane using horizontal and vertical distances from the origin (x and y).
Q: How do I convert polar coordinates to rectangular coordinates?
A: To convert polar coordinates to rectangular coordinates, 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 distance from the origin to the point, and θ is the angle between the positive x-axis and the line segment connecting the origin to the point.
Q: What if the angle θ is negative?
A: If the angle θ is negative, 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 distance from the origin to the point, and θ is the negative angle between the positive x-axis and the line segment connecting the origin to the point.
Q: How do I evaluate the expressions x = r cos(θ) and y = r sin(θ)?
A: To evaluate these expressions, you need to use the values of cosine and sine for the angle θ. You can use a calculator or a trigonometric table to find the values of cosine and sine.
Q: Can I use a calculator to convert polar coordinates to rectangular coordinates?
A: Yes, you can use a calculator to convert polar coordinates to rectangular coordinates. Most calculators have a built-in function to convert polar coordinates to rectangular coordinates.
Q: What if I have a complex number in polar coordinates?
A: If you have a complex number in polar coordinates, you can convert it to rectangular coordinates using the following formulas:
x = r cos(θ) y = r sin(θ)
where x and y are the rectangular coordinates, r is the distance from the origin to the point, and θ is the angle between the positive x-axis and the line segment connecting the origin to the point.
Q: Can I use a computer program to convert polar coordinates to rectangular coordinates?
A: Yes, you can use a computer program to convert polar coordinates to rectangular coordinates. Most programming languages have a built-in function to convert polar coordinates to rectangular coordinates.
Q: What are some common mistakes to avoid when converting polar coordinates to rectangular coordinates?
A: Some common mistakes to avoid when converting polar coordinates to rectangular coordinates include:
- Forgetting to use the correct formula for converting polar coordinates to rectangular coordinates
- Using the wrong value for the angle θ
- Not rationalizing the denominator
- Not simplifying the expressions
Conclusion
In this article, we have answered some of the most frequently asked questions about converting polar coordinates to rectangular coordinates. We have also provided some tips and tricks for avoiding common mistakes when converting polar coordinates to rectangular coordinates.
Key Takeaways
- Polar coordinates are a way of representing the position of a point in a two-dimensional plane using a distance from the origin (r) and an angle from the positive x-axis (θ).
- Rectangular coordinates are a way of representing the position of a point in a two-dimensional plane using horizontal and vertical distances from the origin (x and y).
- To convert polar coordinates to rectangular coordinates, you can use the following formulas: x = r cos(θ) and y = r sin(θ).
- You can use a calculator or a computer program to convert polar coordinates to rectangular coordinates.
- You should avoid common mistakes such as forgetting to use the correct formula, using the wrong value for the angle θ, not rationalizing the denominator, and not simplifying the expressions.
Practice Problems
- Convert the polar coordinates (3, π/4) to rectangular coordinates.
- Convert the polar coordinates (4, -Ï€/3) to rectangular coordinates.
- Convert the polar coordinates (2, 3Ï€/2) to rectangular coordinates.
Answer Key
- (3/√2, 3/√2)
- (-4√3/2, -4/2)
- (0, -2)
References
- "Polar Coordinates" by Math Open Reference
- "Rectangular Coordinates" by Math Open Reference
- "Converting Polar Coordinates to Rectangular Coordinates" by Wolfram MathWorld