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Trigonometry - Polar Coordinates and Complex Numbers

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    Study Guide Trigonometry – Polar Coordinates and Complex Numbers 1. Geometry of Complex Numbers Complex numbers aren’t just abstract symbols — they can be drawn, measured, and visualized . In this chapter, we’ll explore how complex numbers work geometrically and how to move between rectangular and polar forms. 1. 1 What Is a Complex Number? A complex number can be written in the form • (a) and (b) are real numbers • (i) is the imaginary unit, where (i 2 = - 1) Every complex number represents a point in a plane , called the complex plane . 1.2 The Complex Plane The complex plane looks like a regular coordinate plane, but with a special meaning: • The x - axis is called the real axis • The y - axis is called the imaginary axis A complex number (a + b i ) is plotted as the point (a, b) Example 1 : Plotting Complex Numbers Plot the following complex numbers:

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    Study Guide Figure 1 Complex numbers plotted in the complex plane. • (4 - 2i) • ( - 3 + 2i) • ( - 5 - 3i) Each number becomes a point: • (4 - 2i → (4, - 2)) • ( - 3 + 2i → ( - 3, 2)) • ( - 5 - 3i → ( - 5, - 3)) These points are placed in the complex plane just like coordinates in algebra. 1.3 Polar Form of Complex Numbers Instead of using coordinates ((x, y)), we can describe a complex number using: • Distance from the origin (called the modulus ) • Angle from the positive x - axis (called the argument or amplitude )

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    Study Guide The Modulus (Absolute Value) For a complex number The modulus is This is simply the distance from the origin to the point ((x, y)). The Argument (Angle) The argument of a complex number is the angle ( θ) formed between: • the positive x - axis • the line connecting the origin to the point 1.4 Writing Complex Numbers in Polar Form Using trigonometry: • (x = r cos θ) • (y = r sin θ) So a complex number can be written as: This is called the polar form of a complex number. Sometimes you’ll see: written as cis ( θ ) .

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    Study Guide 1.5 Conjugate of a Complex Number If Then the conjugate of (z) is: The conjugate reflects the point across the real axis. Example 2 : Converting to Polar Form Convert (5 - 3i) to polar form. Figure 2 Drawing for Example 2. Step 1: Find the modulus Step 2: Find the reference angle This gives a reference angle of about (31 ° ).

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    Study Guide Step 3: Adjust for the quadrant Since (5 - 3i) lies in the fourth quadrant : Final Answer 1.6 Multiplying Complex Numbers in Polar Form To multiply two complex numbers: • Multiply their moduli • Add their angles If and Then: Example 3 : Product Formula This rule comes from expanding the expressions and using trigonometric identities. The result always simplifies nicely into another polar form. 1 . 7 Dividing Complex Numbers in Polar Form To divide two complex numbers: • Divide their moduli • Subtract their angles

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    Study Guide If and Then: Example 4 : Quotient Formula If and find the quotient Step 1: Write the Quotient Start by dividing: Factor out constants:

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    Study Guide Step 2: Multiply by the Conjugate To simplify the denominator, multiply by the conjugate of So we multiply top and bottom by this expression. Step 3: Simplify the Denominator This is a difference of squares: And we know the identity: So the denominator becomes: That makes things much simpler! Step 4: Multiply the Numerator Now multiply: Distribute carefully: Since ( i 2 = - 1 ), this becomes:

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    Study Guide Step 5: Recognize the Trig Identities Use these identities: So the expression becomes: Final Answer Example 5 : Putting It All Together Let: Product Quotient Since negative angles can be rewritten:

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