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Multiply and Divide in Polar Form
If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divide: r1/r2 and subtract arguments. Meaning: lengths multiply; angles add. That is why polar form beats expanding foil for powers and roots.
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Multiply and Divide in Polar Form
What is it?
If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divide: r1/r2 and subtract arguments. Meaning: lengths multiply; angles add. That is why polar form beats expanding foil for powers and roots.
In simple words
- If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)).
- Divide: r1/r2 and subtract arguments.
- Meaning: lengths multiply; angles add.
- Add arguments, then bring back into the book's principal interval.
Important terms
MULTIPLY AND DIVIDE IN POLAR FORM: If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divide: r1/r2 and subtract arguments. Meaning: lengths multiply; angles add. That is why polar form beats expanding foil for powers and roots.
1. PRODUCT: Draw two arrows: the product is longer or shorter by the extra factor, and turned extra. |z1 z2| = |z1||z2| matches. Arg(z1 z2) = Arg z1 + Arg z2, then reduce to the principal interval.
2. QUOTIENT: z1/z2 = (r1/r2) (cos(t1-t2)+i sin(t1-t2)). Inverse: 1z = (1r) (cos(-t)+i sin(-t)). Check z2 != 0.
What you should learn
- State the main ideas and notation used in multiply and divide in polar form.
- Apply the rules of multiply and divide in polar form in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- i * i = -1 (pi/2 + pi/2 = pi)
- r1 r2
- Reduce by 2 pi if needed
PRODUCT
Draw two arrows: the product is longer or shorter by the extra factor, and turned extra. |z1 z2| = |z1||z2| matches. Arg(z1 z2) = Arg z1 + Arg z2, then reduce to the principal interval.
QUOTIENT
z1/z2 = (r1/r2) (cos(t1-t2)+i sin(t1-t2)). Inverse: 1z = (1r) (cos(-t)+i sin(-t)). Check z2 != 0.
Recommended for Class 11
Full school-level explanation
If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divide: r1/r2 and subtract arguments. Meaning: lengths multiply; angles add. That is why polar form beats expanding foil for powers and roots.
Times, then divide:
WHAT — MULTIPLY AND DIVIDE IN POLAR FORM
If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divide: r1/r2 and subtract arguments. Meaning: lengths multiply; angles add. That is why polar form beats expanding foil for powers and roots.
RULE — 1. PRODUCT
Draw two arrows: the product is longer or shorter by the extra factor, and turned extra. |z1 z2| = |z1||z2| matches. Arg(z1 z2) = Arg z1 + Arg z2, then reduce to the principal interval.
METHOD — 2. QUOTIENT
z1/z2 = (r1/r2) (cos(t1-t2)+i sin(t1-t2)). Inverse: 1z = (1r) (cos(-t)+i sin(-t)). Check z2 != 0.
WORKED EXAMPLE
Worked: [sqrt(2) cis pi/4]^2. r^2 = 2 2 theta = pi/2 = 2 (cos pi/2 + i sin pi/2) = 2i check: (1+i)^2 = 1 + 2i + i^2 = 2i
PRODUCT
Draw two arrows: the product is longer or shorter by the extra factor, and turned extra. |z1 z2| = |z1||z2| matches. Arg(z1 z2) = Arg z1 + Arg z2, then reduce to the principal interval.
Remember
- add the turns
Examples
- i * i = -1 (pi/2 + pi/2 = pi)
- r1 r2
- Reduce by 2 pi if needed
QUOTIENT
z1/z2 = (r1/r2) (cos(t1-t2)+i sin(t1-t2)). Inverse: 1z = (1r) (cos(-t)+i sin(-t)). Check z2 != 0.
Remember
- subtract the turns
Examples
- 1i = -i
- Args subtract
- r becomes 1r
How to answer a multiply and divide in polar form question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- multiply: r's multiply, args add
- divide: r's divide, args subtract
- 2i
- reduce Arg
- add the turns
- subtract the turns
- multiply: r's multiply, args add
- divide: r's divide, args subtract
How the ideas connect
Worked: [sqrt(2) cis pi/4]^2. r^2 = 2 2 theta = pi/2 = 2 (cos pi/2 + i sin pi/2) = 2i check: (1+i)^2 = 1 + 2i + i^2 = 2i
Deeper understanding
Go beyond memorising Multiply and Divide in Polar Form. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: [sqrt(2) cis pi/4]^2. r^2 = 2 2 theta = pi/2 = 2 (cos pi/2 + i sin pi/2) = 2i check: (1+i)^2 = 1 + 2i + i^2 = 2i
Common mistakes
Exam focus
Core Idea
1. PRODUCT
2. QUOTIENT
Worked Example
Exam focus
Quick revision
- multiply: r's multiply, args add
- divide: r's divide, args subtract
- 2i
- reduce Arg
- MULTIPLY AND DIVIDE IN POLAR FORM: If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i s
- 1. PRODUCT: Draw two arrows: the product is longer or shorter by the extra factor, and turned extra. |z1 z2| = |
Add arguments, then bring back into the book's principal interval.
For long answers on Multiply and Divide in Polar Form: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- multiply: r's multiply, args add
- divide: r's divide, args subtract
- 2i
- reduce Arg
Important formulas
- multiply: r's multiply, args add
- divide: r's divide, args subtract
- 2i
Common exam traps
- Add arguments, then bring back into the book's principal interval.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Add arguments, then bring back into the book's principal interval.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Add arguments, then bring back into the book's principal interval.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
multiply: r's multiply, args add
divide: r's divide, args subtract
2i
reduce Arg
If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divide: r1/r2 and subtract arguments. Meaning: l
- Definition
- If z1 = r1 (cos t1 + i sin t1) and z2 = r2 (cos t2 + i sin t2), then z1 z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divide: r1/r2 and subtract arguments. Meaning: l
- Example
- i * i = -1 (pi/2 + pi/2 = pi)
- Critical warning
- Add arguments, then bring back into the book's principal interval.
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Exam Practice
Define
Q1. Define Multiply and Divide in Polar Form.
Explain
Q2. Explain: PRODUCT.
Example
Q3. Give one clear example of Multiply and Divide in Polar Form and say why it fits.
Mistake
Q4. State one common mistake students make about Multiply and Divide in Polar Form.
Exam note
Q5. Write a short exam note on Multiply and Divide in Polar Form: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.