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How to Convert Polar Coordinates to Rectangular Form

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Step-by-Step Example: Converting Polar Coordinates to x and y

Learning Polar To Rectangular conversion helps you switch between different coordinate systems used in school math, board exams, and science applications. Mastering this makes it easy to plot points, solve geometry questions, and handle competitive exam problems where both polar and rectangular forms appear.


Formula Used in Polar To Rectangular

The standard formula is: \( x = r \cos \theta, \ y = r \sin \theta \)


Here’s a helpful table to understand Polar To Rectangular more clearly:


Polar To Rectangular Table

Coordinate TypeRepresentationFormula Used
Polar (r, θ)
Rectangular (x, y) \(x = r \cos \theta\)
\(y = r \sin \theta\)

This table shows how the pattern of Polar To Rectangular conversion is used when switching between coordinate systems in geometry and physics.


Worked Example – Solving a Problem

1. A point has polar coordinates \( (4, \frac{\pi}{3}) \). Find its rectangular coordinates.
Step 1: Identify each value: \( r = 4 \), \( \theta = \frac{\pi}{3} \).

Step 2: Use the formulas: \( x = r \cos \theta \), \( y = r \sin \theta \).

Step 3: Calculate \( x \):
 \( x = 4 \times \cos \frac{\pi}{3} = 4 \times \frac{1}{2} = 2 \)

Step 4: Calculate \( y \):
 \( y = 4 \times \sin \frac{\pi}{3} = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \)

Step 5: State the result:
The rectangular coordinates are (2, \( 2\sqrt{3} \)).


2. Convert polar coordinates \( (5, 45^\circ) \) to rectangular form.
Step 1: \( r = 5, \ \theta = 45^\circ = \frac{\pi}{4} \) radians.

Step 2: \( x = 5 \cos 45^\circ = 5 \times \frac{1}{\sqrt{2}} = \frac{5}{\sqrt{2}} \)

Step 3: \( y = 5 \sin 45^\circ = 5 \times \frac{1}{\sqrt{2}} = \frac{5}{\sqrt{2}} \)

Step 4: Rectangular coordinates: \( (\frac{5}{\sqrt{2}}, \frac{5}{\sqrt{2}}) \)

Practice Problems

  • Convert the polar coordinates \( (3, \frac{\pi}{2}) \) to rectangular coordinates.
  • If a point has polar coordinates \( (7,0) \), what are its rectangular coordinates?
  • Find rectangular coordinates of \( (10, 120^\circ) \).
  • Given polar points \( (2, \frac{\pi}{4}) \) and \( (4, \pi) \), write both in rectangular form.

Common Mistakes to Avoid

  • Mixing up angle measurements—always check if θ is in degrees or radians.
  • Forgetting to use calculators in radian mode when θ is in radians.
  • Applying the formulas incorrectly (like swapping sine and cosine).
  • Missing negative signs when points are in certain quadrants.

Real-World Applications

The concept of Polar To Rectangular conversion appears in engineering, navigation, computer graphics, and robotics. From plotting the location of stars to analyzing circuits and designing 2D animations, these transformations are essential. Vedantu helps students practice these skills for practical and exam success.


Related Concepts and Further Study

To understand more about these systems, check out Polar Coordinates and Cartesian Coordinates. For a complete view of coordinate systems, see Coordinate Geometry. Interested in the applications of these conversions in complex numbers? Explore Polar Form of Complex Numbers.


We explored the idea of Polar To Rectangular, how to use formulas, solve problems, and apply it in real life. Keep practicing these conversions with Vedantu to gain speed and accuracy in your school and competitive exams!


FAQs on How to Convert Polar Coordinates to Rectangular Form

1. How do you convert from polar to rectangular coordinates?

To convert **polar coordinates** (r, θ) to **rectangular coordinates** (x, y), use the following formulas:
x = r × cos(θ)
y = r × sin(θ)
Here, **r** is the radius (distance from origin), and **θ** is the angle in degrees or radians. This method is commonly used in geometry, trigonometry, and physics to find **xy-coordinates** from **polar form**.

2. How do you convert the polar coordinates 6120 to rectangular form?

If the **polar coordinates** are **(6, 120°)**, you can convert to rectangular as follows:
x = 6 × cos(120°) = 6 × (-0.5) = -3
y = 6 × sin(120°) = 6 × (√3/2) ≈ 6 × 0.866 = 5.196
Therefore, the **rectangular coordinates** are (-3, 5.196).

3. How do you convert polar coordinates to xy coordinates?

To convert **polar coordinates (r, θ)** to **xy coordinates**, use:
x = r × cos(θ)
y = r × sin(θ)
Make sure the angle **θ** is in degrees or radians as required by your calculator or method.

4. What is the equation for rectangular to polar converter?

To convert **rectangular coordinates (x, y)** to **polar coordinates (r, θ)**, use:
r = √(x² + y²)
θ = tan⁻¹(y/x)
These formulas help in changing from the **Cartesian (xy)** system to the **Polar (rθ)** system.

5. What is the polar to rectangular coordinates formula?

The **polar to rectangular conversion formulas** are:
x = r × cos(θ)
y = r × sin(θ)
Where **r** = radius and **θ** = angle.

6. How do you use a polar to rectangular calculator?

To use a **polar to rectangular calculator**:
1. Enter the value of **r** (radius) and **θ** (angle).
2. The calculator will use x = r × cos(θ) and y = r × sin(θ) to give the **x** and **y** values.
3. Check if the angle is in degrees or radians, as this affects the calculation.

7. What is the polar to rectangular form of complex numbers?

A **complex number** in **polar form** is written as r(cosθ + i sinθ). Its **rectangular form** is x + iy, where:
x = r × cos(θ)
y = r × sin(θ)
This conversion is important in algebra and circuits to switch between forms.

8. How is polar to rectangular conversion used in circuits?

In **AC circuits** and **phasor analysis**, quantities like voltage and current are often given in **polar form** (magnitude and phase). To perform calculations, convert to **rectangular form** using x = r × cos(θ) and y = r × sin(θ), where real part represents magnitude and imaginary part represents phase shift.

9. How do you convert rectangular to polar without a calculator?

To convert from **rectangular (x, y)** to **polar (r, θ)** without a calculator:
• Find r = √(x² + y²) manually.
• Calculate θ = tan⁻¹(y/x) using trigonometric tables or basic ratios.
If angles are standard (like 0°, 45°, 90°, 180°), they can be quickly deduced without calculation.

10. What is a polar to rectangular equation calculator?

A **polar to rectangular equation calculator** is a digital or online tool that converts values in **polar coordinates** into **rectangular coordinates** using the formulas x = r × cos(θ) and y = r × sin(θ). This is helpful for physics, maths, and engineering problems.

11. What is the difference between polar and rectangular coordinates?

The key difference is:
• **Polar coordinates (r, θ)** use a distance and an angle from the origin.
• **Rectangular coordinates (x, y)** use horizontal and vertical distances from the origin.
The two are related by trigonometric conversion formulas.

12. How to convert polar to rectangular on a Casio calculator?

On many **Casio calculators**:
1. Press the MODE key to select **complex number** calculations.
2. Enter the polar form using the format **(r ∠ θ)**.
3. Use the calculator's built-in function (often labelled with a '→Rec' or similar button) to convert to **rectangular form**.
Refer to your model’s manual for exact instructions.