Grade 11 · Complex polar form · 455 of 600
Complex polar form · Modulus r=0.5; Final angle (degrees)=-45
Evaluate at the signed angle θ=-27° (three fifths of that final angle). Find its real and imaginary components and verify its modulus. Givens: Modulus r=0.5; Final angle (degrees)=-45.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A vector has length r=0.5; its angle is measured counterclockwise from the positive horizontal axis. The animation's final angle is -45°. Evaluate at the signed angle θ=-27° (three fifths of that final angle). Find its real and imaginary components and verify its modulus.
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A starting point
Use (r cos θ,r sin θ) with degrees. The dot product with (1,0) selects the first component.
Work through the reasoning
Step 1
Identify the model and target
A vector has length r=0.5; its angle is measured counterclockwise from the positive horizontal axis. The animation's final angle is -45°. Evaluate at the signed angle θ=-27° (three fifths of that final angle). The governing relation is z=r cos θ+i r sin θ; |z|=r. Use (r cos θ,r sin θ) with degrees. The dot product with (1,0) selects the first component.
Step 2
Substitute and calculate
At θ=-27°, the coordinates are (0.5 cos(-27°), 0.5 sin(-27°))=(0.445503,-0.226995). Their squared sum is 0.25, and the dot product with (1,0) selects the first coordinate.
Step 3
Check the mathematical meaning
The squared components sum to r²=0.25. Their signs must match the quadrant of θ=-27°. A negative angle rotates clockwise. Angle in degrees: -27; Real part: 0.446; Imaginary part: -0.227; Modulus: 0.5. Decimal values are rounded, so use unrounded intermediate values.
The answer
At θ=-27°, the coordinates are (0.5 cos(-27°), 0.5 sin(-27°))=(0.445503,-0.226995). Their squared sum is 0.25, and the dot product with (1,0) selects the first coordinate. The squared components sum to r²=0.25. Their signs must match the quadrant of θ=-27°. A negative angle rotates clockwise. Animation check: Angle in degrees: -27; Real part: 0.446; Imaginary part: -0.227; Modulus: 0.5. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. A vector has length r=0.5; its angle is measured counterclockwise from the positive horizontal axis. The animation's final angle is -45°. Evaluate at the signed angle θ=-27° (three fifths of that final angle). The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
z=r cos θ+i r sin θ; |z|=r
03 · Reflect and transfer
Explain what changes and why.
Increase the angle by 180°. What changes sign, and which quantity stays unchanged?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.