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Grade 11 · Complex polar form

Complex polar form: A second-quadrant long radius

Complex polar form: investigate a second-quadrant long radius with modulus r = 2.5; final angle (degrees) = 150.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Complex polar form: A second-quadrant long radius. Angle in degrees: 0. Real part: 2.5. Imaginary part: 0. Modulus: 2.5Real and imaginary componentsxy0r = 2.5θ = 0°Equal axis scales · coordinates in the readouts
The complex plane uses equal axis scales. Angles are selected in degrees and converted to radians internally. This is multiplication by a unit complex exponential; no addition or change of modulus is implied.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Angle in degrees
0
Real part
2.5
Imaginary part
0
Modulus
2.5

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

Polar form describes a complex number with a length and angle. Rotating changes the signed real and imaginary components but preserves their squared sum. Reading both coordinates together avoids losing quadrant information when converting between polar and rectangular descriptions. This investigation starts with Modulus r = 2.5; Final angle (degrees) = 150. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

z=r cos θ+i r sin θ; |z|=r

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Start at the positive real number r. Distinguish the radius from either individual coordinate. The starting case is “A second-quadrant long radius.”

  2. STEP 2

    Follow the changing quantity

    Rotate toward the selected angle. Positive angles turn counterclockwise and negative angles clockwise.

  3. STEP 3

    Explain and test the result

    Square and add the two displayed components to check that their sum is r², including at axis crossings.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Modulus r = 2.5; Final angle (degrees) = 150. Pause the timeline at 80%. Given angle in degrees = 120, calculate real part, imaginary part, modulus. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At θ=120°, the coordinates are (2.5 cos(120°), 2.5 sin(120°))=(-1.25,2.165064). Their squared sum is 6.25, and the dot product with (1,0) selects the first coordinate. Results: Real part: -1.25; Imaginary part: 2.165; Modulus: 2.5. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Algebra

Describe a complex number by magnitude and angle

Choose an argument in the correct quadrant.