Imaginary Numbers Polar to Rectangular Calculator

Magnitude r

Angle theta

Angle unit

Angle direction

Decimal places

Advanced option

Normalize angle before display

Reset

Formula Used

Polar complex form is written as:

z = r(cos theta + i sin theta)

The rectangular form is:

z = x + yi

Use these conversion formulas:

x = r cos theta

y = r sin theta

If the angle is in degrees, the calculator first uses:

theta radians = theta degrees × pi / 180

For clockwise angles, theta is treated as negative before conversion.

How to Use This Calculator

Enter the magnitude, also called radius r.
Enter the polar angle theta.
Select degrees or radians.
Choose the angle direction.
Set decimal places for rounded output.
Press Calculate to see the result above the form.
Use CSV or PDF buttons to save the output.

Example Data Table

Magnitude r	Angle	Unit	Direction	Rectangular Form
1	90	Degrees	Counterclockwise	0 + 1i
2	30	Degrees	Counterclockwise	1.732 + 1i
5	60	Degrees	Counterclockwise	2.5 + 4.330i
4	45	Degrees	Clockwise	2.828 - 2.828i

Understanding Polar Complex Conversion

Imaginary numbers often appear in two useful forms. Rectangular form shows a complex number as a real part and an imaginary part. Polar form shows the same value through a radius and an angle. This calculator connects both views. It turns r and theta into a plus bi, so the number becomes easier to plot, compare, or use in later work.

Why Polar Form Matters

Polar form is natural when size and direction matter. It appears in vectors, electrical phase work, signals, rotations, and geometry. The radius tells the distance from the origin. The angle tells the direction from the positive real axis. Together, they locate one point on the complex plane.

How Rectangular Form Helps

Rectangular form is practical for addition, subtraction, graphing, and many algebra steps. The real part moves left or right. The imaginary part moves up or down. Once the calculator finds both parts, you can place the point on axes without estimating the triangle.

Accuracy and Units

The angle may be entered in degrees or radians. Degrees are common in classroom problems. Radians are common in calculus, physics, and engineering. The calculator converts degrees internally before using the trigonometric formulas. Precision control lets you round results for reports, homework, or quick checks.

Advanced Use Cases

Use the tool to check roots, phase angles, impedance values, and rotating vectors. Try angles beyond 360 degrees. The sine and cosine functions still return the matching rectangular position. A negative angle also works, because it turns clockwise instead of counterclockwise.

Reading the Result

The output shows x, y, and the final complex expression. If y is positive, the expression uses plus bi. If y is negative, it uses minus bi. The magnitude check confirms the distance. The angle check helps verify the direction. These checks make the answer easier to trust.

Best Practice

Keep enough decimal places during calculation. Round only the final answer when possible. For exact special angles, compare the rounded value with known sine and cosine values. This habit prevents small rounding changes from causing confusion in larger problems.

For shared work, download the result and attach it with notes. Saved outputs help teams review assumptions, units, and rounding choices later clearly.

FAQs

What is polar form for an imaginary number?

Polar form writes a complex number using radius and angle. It is usually shown as r(cos theta + i sin theta). The radius gives distance from the origin. The angle gives direction on the complex plane.

What is rectangular form?

Rectangular form writes the complex number as a + bi. The a value is the real part. The b value is the imaginary coefficient. It is useful for graphing and basic algebra operations.

Can I enter angles in radians?

Yes. Select radians from the angle unit field. The calculator will use the value directly in sine and cosine formulas. Choose degrees only when your angle is measured in degrees.

What does clockwise direction mean?

Clockwise direction treats the entered angle as negative. This follows the standard complex plane rule. Positive angles rotate counterclockwise, while negative angles rotate clockwise from the positive real axis.

Why is the result sometimes very close to zero?

Trigonometric calculations can create tiny decimal leftovers. For example, cos 90 degrees may produce a very small number instead of exact zero. The calculator rounds tiny values for cleaner output.

What does the magnitude check show?

The magnitude check recalculates distance from the rectangular parts. It uses the square root of x squared plus y squared. It should match the entered radius after normal rounding.

When should I use more decimal places?

Use more decimal places for engineering, signals, impedance, and detailed homework. Extra precision reduces rounding error. You can use fewer decimals for quick estimates or simple classroom examples.

Can I download the result?

Yes. After calculating, use the CSV button for spreadsheet data. Use the PDF button for a simple report. Both options save the main input and output values.