MathematicsChapter 1: Complex Numbers
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)).
Learning objectives
- 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.
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: lengths multiply; angles add. That is why polar form beats expanding foil for powers and roots.
Mathematical form
z1 = r1 (cos t1 + i sin t1) and z2
z2 = r1 r2 (cos(t1+t2) + i sin(t1+t2)). Divid
theta = pi/2
Where / symbols
- 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: 1/z = (1/r) (cos(-t)+i sin(-t)). Check z2 != 0.
Overview
Times, then divide:
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.
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.
Examples 1/i = -i · Args subtract · r becomes 1/r
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 + i2 = 2i
How to answer a multiply and divide in polar form question
- 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.
Worked example
- [sqrt(2) cis pi/4]^2.
- r2 = 2
- 2 theta = pi/2
- = 2 (cos pi/2 + i sin pi/2) = 2i
- check: (1+i)2 = 1 + 2i + i2 = 2i
In simple words
Add arguments, then bring back into the book's principal interval.
Key points
- multiply: r's multiply, args add
- divide: r's divide, args subtract
- add the turns
- subtract the turns
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Multiply and Divide in Polar Form
Sticky Notes · Quick Revision
Core Idea
- 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.
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.
Worked Example
- Worked: [sqrt(2) cis pi/4]^2.
- r2 = 2
- 2 theta = pi/2
- = 2 (cos pi/2 + i sin pi/2) = 2i
- check: (1+i)2 = 1 + 2i + i2 = 2i
Common Misconception
- Add arguments, then bring back into the book's principal interval.
FBISE Exam Check
- State the rule for multiply and divide in polar form before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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