Convert The Polar Coordinates { \left(4 \sqrt{2}, \frac{3 \pi}{4}\right)$}$ Into Rectangular Form. Express Your Answer In Simplest Radical Form.

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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 explore how to convert polar coordinates to rectangular form, with a focus on the given polar coordinates {\left(4 \sqrt{2}, \frac{3 \pi}{4}\right)$}$.

Understanding Polar Coordinates

Polar coordinates are represented as (r, θ), where r is the distance from the origin to the point, and θ is the angle from the positive x-axis to the line connecting the origin to the point. In the given polar coordinates {\left(4 \sqrt{2}, \frac{3 \pi}{4}\right)$}$, the distance from the origin to the point is 424 \sqrt{2}, and the angle from the positive x-axis to the line connecting the origin to the point is 3π4\frac{3 \pi}{4}.

Converting Polar Coordinates to Rectangular Form

To convert polar coordinates to rectangular form, we 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 from the positive x-axis to the line connecting the origin to the point.

Applying the Formulas

Using the given polar coordinates {\left(4 \sqrt{2}, \frac{3 \pi}{4}\right)$}$, we can apply the formulas to find the rectangular coordinates:

x = 424 \sqrt{2} cos(3Ï€4\frac{3 \pi}{4}) y = 424 \sqrt{2} sin(3Ï€4\frac{3 \pi}{4})

Evaluating the Trigonometric Functions

To evaluate the trigonometric functions, we need to know the values of cos(3Ï€4\frac{3 \pi}{4}) and sin(3Ï€4\frac{3 \pi}{4}). Using the unit circle or a trigonometric table, we can find that:

cos(3Ï€4\frac{3 \pi}{4}) = -12\frac{1}{\sqrt{2}} sin(3Ï€4\frac{3 \pi}{4}) = 12\frac{1}{\sqrt{2}}

Substituting the Values

Substituting the values of cos(3Ï€4\frac{3 \pi}{4}) and sin(3Ï€4\frac{3 \pi}{4}) into the formulas, we get:

x = 424 \sqrt{2} (-12\frac{1}{\sqrt{2}}) y = 424 \sqrt{2} (12\frac{1}{\sqrt{2}})

Simplifying the Expressions

Simplifying the expressions, we get:

x = -44 y = 424 \sqrt{2}

Conclusion

In this article, we have explored how to convert polar coordinates to rectangular form, with a focus on the given polar coordinates {\left(4 \sqrt{2}, \frac{3 \pi}{4}\right)$}$. We have applied the formulas x = r cos(θ) and y = r sin(θ) to find the rectangular coordinates, and have simplified the expressions to obtain the final answer.

Rectangular Coordinates

The rectangular coordinates of the given polar coordinates {\left(4 \sqrt{2}, \frac{3 \pi}{4}\right)$}$ are:

x = -44 y = 424 \sqrt{2}

Simplifying Radical Form

To express the answer in simplest radical form, we can simplify the expression for y:

y = 424 \sqrt{2}

This expression is already in simplest radical form.

Final Answer

The final answer is:

Introduction

In our previous article, we explored how to convert polar coordinates to rectangular form, with a focus on the given polar coordinates {\left(4 \sqrt{2}, \frac{3 \pi}{4}\right)$}$. In this article, we will answer some frequently asked questions about converting polar coordinates to rectangular form.

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).

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.

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 distance from the origin to the point, and θ is the angle from the positive x-axis to the line connecting the origin to the point.

Q: What if the angle is in radians?

A: If the angle is in radians, you can use the same formulas as above. However, if the angle is in degrees, you will need to convert it to radians first.

Q: Can I use a calculator to convert polar coordinates to rectangular form?

A: Yes, you can use a calculator to convert polar coordinates to rectangular form. Most calculators have a built-in function for converting polar coordinates to rectangular form.

Q: What if the distance is a negative number?

A: If the distance is a negative number, you will need to adjust the angle accordingly. For example, if the distance is -r, you will need to add π to the angle.

Q: Can I convert polar coordinates to rectangular form using a graphing calculator?

A: Yes, you can convert polar coordinates to rectangular form using a graphing calculator. Most graphing calculators have a built-in function for converting polar coordinates to rectangular form.

Q: What if I have a polar equation and I want to convert it to rectangular form?

A: If you have a polar equation and you want to convert it to rectangular form, you can use the following steps:

  1. Convert the polar equation to a parametric equation.
  2. Use the parametric equation to find the rectangular coordinates.

Q: Can I use a computer program to convert polar coordinates to rectangular form?

A: Yes, you can use a computer program to convert polar coordinates to rectangular form. Most computer programs have a built-in function for converting polar coordinates to rectangular form.

Conclusion

In this article, we have answered some frequently asked questions about converting polar coordinates to rectangular form. We hope that this article has been helpful in clarifying any confusion you may have had about this topic.

Additional Resources

If you are interested in learning more about converting polar coordinates to rectangular form, we recommend checking out the following resources:

  • A calculus textbook or online resource
  • A graphing calculator or computer program
  • A math website or online community

Final Answer

The final answer is:

x = -44 y = 424 \sqrt{2}