Multiply and divide polar forms by combining moduli and angles
Multiply moduli and add arguments, or divide moduli and subtract arguments, then normalise the result and check its Cartesian direction.
Exam scope and assessment notes
Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.9. Paper 1 algebra and radian trigonometry are assumed. Use Cartesian working for products, quotients and the two square roots of a non-zero complex number. Arguments usually use −π < θ ≤ π; an alternative interval 0 ≤ θ < 2π is acceptable unless specified. The argument of zero is undefined. General nth-root methods are beyond this mapped sequence. Written practice is self-checked; human subject-teacher review remains pending.
Before you begin
- Convert between Cartesian and polar forms.
- Use radian angle sums and exact trigonometric values.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the modulus of a product with factor moduli 2 and 3?
What you will learn
- Calculate polar products and quotients with normalised arguments.
- Interpret a non-zero multiplier as a scale factor and rotation.
Products add directions and multiply lengths
For z₁ = r₁cis θ₁ and z₂ = r₂cis θ₂, their product is r₁r₂cis(θ₁ + θ₂). Expanding the Cartesian product and using the sine and cosine angle-addition identities proves this rule. Thus |z₁z₂| = |z₁||z₂| and the product’s argument is the sum modulo 2π, provided both factors are non-zero. Multiplication by zero gives zero, whose argument remains undefined.
For 2cis(2π/3) multiplied by 3cis(5π/6), the modulus is 6 and the angle sum is 3π/2. In the usual principal interval this becomes −π/2, so the product is −6i. Normalising the sum changes its representative angle, not the complex number. Multiplying by a fixed non-zero number scales every distance from the origin by its modulus and rotates anticlockwise through its argument, with negative angles indicating clockwise rotation.
Quotients subtract directions
If z₂ ≠ 0, z₁/z₂ = (r₁/r₂)cis(θ₁ − θ₂). For the same two non-zero values, the quotient is (2/3)cis(−π/6) = √3/3 − i/3. Its modulus is less than one because the numerator modulus is smaller. Division by a multiplier reverses its rotation and uses the reciprocal scale factor; it does not subtract the two moduli.
Keep exact angles through the calculation and reduce the final angle to the requested interval. For −19π/12, adding 2π gives 5π/12. A result expressed with −19π/12 is the same direction, but it does not satisfy the usual principal interval. If the numerator is zero and the denominator non-zero, the quotient is zero and no resulting argument can be assigned.
Pause and explain
Which principal angle represents 3π/2?
Worked example
Find the product and quotient of z₁ = 2cis(2π/3) and z₂ = 3cis(5π/6), giving principal arguments.
Show the worked solution
- The product modulus is 6 and its angle is 3π/2, equivalent to −π/2.
- The quotient modulus is 2/3 and its angle is 2π/3 − 5π/6 = −π/6.
- Cartesian checks give product −6i and quotient √3/3 − i/3.
Answer Product −6i, argument −π/2; quotient √3/3 − i/3, argument −π/6.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Multiply 2cis(π/4) by √2cis(π/6). Give polar and exact Cartesian forms.
Give me a hint
Add the angles, then use angle-addition values for 5π/12.
Compare my reasoning
- The product is 2√2cis(5π/12).
- cos(5π/12) = (√6 − √2)/4 and sin(5π/12) = (√6 + √2)/4.
- Multiplying these by 2√2 gives (√3 − 1) + i(√3 + 1).
2√2cis(5π/12) = (√3 − 1) + i(√3 + 1).
Look for these in your work
- Multiply moduli and add the exact angles.
- Check both exact Cartesian components.
Divide 4cis(−5π/6) by 2cis(3π/4), using the principal argument.
Give me a hint
Subtract the denominator angle before normalising the result.
Compare my reasoning
- The modulus is 4/2 = 2.
- The raw angle is −5π/6 − 3π/4 = −19π/12.
- Add 2π to obtain 5π/12 in the principal interval.
2cis(5π/12), principal argument 5π/12.
Look for these in your work
- Preserve the order of the quotient’s angle subtraction.
- Normalise the exact angle to the requested interval.
A supplied transformation multiplies by q = 2cis(−π/3). Apply it to z = 3cis(5π/6), state the scale and rotation and explain how to undo it.
Give me a hint
The multiplier controls the scale and rotation about the origin.
Compare my reasoning
- The transformed modulus is 6 and its angle is 5π/6 − π/3 = π/2, giving 6i.
- The transformation scales by 2 and rotates clockwise by π/3 about the origin.
- Dividing by q scales by 1/2 and rotates anticlockwise by π/3, recovering z.
Image 6i; scale 2 and rotation −π/3; undo by dividing by q.
Look for these in your work
- Use the supplied multiplier’s modulus and direction.
- Explain the inverse transformation with reciprocal scale and opposite rotation.
Common mistakes
- Multiply or divide moduli; do not add or subtract them.
- Reduce an angle to the stated interval after combining directions.
- Division needs a non-zero denominator; directional rules need non-zero results.
What can you explain now?
What scale and rotation does division by 2cis(π/3) apply?
Compare with the explanation
Scale 1/2 and rotation −π/3 about the origin.
Division uses the reciprocal modulus and opposite directed angle.
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Make it stick
Calculate a polar product or quotient, normalise its argument and describe the associated rotation and scale about the origin.
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