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Monday 20th of May
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Solving Linear Systems of Equations by Elimination
The Quadratic Formula
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Polar Form of a Complex Number
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Simplifying Complex Fractions
Common Logs
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Multiplying Fractions in General
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Higher Degrees and Variable Exponents
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Writing a Rational Expression in Lowest Terms
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The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
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Solving Equations with One Log Term
Combining Operations
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Graphing Inequalities in Two Variables
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Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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Polar Form of a Complex Number

Example 1

Writing a Complex Number in Polar Form

Write the complex number in polar form.


The absolute value of z is

and the angle θ is given by

Because and because lies in Quadrant III, you choose θ to be θ = π + π/3 = 4π/3. Thus, the polar form is

(See the figure below)

The polar form adapts nicely to multiplication and division of complex numbers. Suppose you are given two complex numbers

z1 = r1(cos θ1 + i sin θ1 ) and z2 = r2(cos θ2 + i sin θ2 )

The product of z1 and z2 is

z1 z2 = r1 r2 cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 )

= r1 r2 [(cos θ1cos θ2 - i sin θ1sin θ2) + i(sin θ1cos θ2 + cos θ1sin θ2)]

Using the sum and difference formulas for cosine and sine, you can rewrite this equation as

z1 z2 = r1 r2 [(cos1+ θ2) + i sin (θ1+ θ2)]

This establishes the first part of the following rule. Try to establish the second part on your own.


Product and Quotient of Two Complex Numbers

Let z1 = r1(cos θ1 + i sin θ1 ) and z2 = r2(cos θ2 + i sin θ2 ) be complex numbers.

z1 z2 = r1 r2 [(cos1+ θ2) + i sin (θ1+ θ2)] Product

Note that this rule says that to multiply two complex numbers you multiply moduli and add arguments, whereas to divide two complex numbers you divide moduli and subtract arguments.


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