# Ch 8  Complex Numbers, Polar Equations, and Parametric Equations

## Ch 8.1  Complex Numbers

Solving x2–10x+34=0;
Simplifying Complex Number in the Form of a+bi

Multiplying (2+3i)(3–4i);
Multiplying (5–2i)2;
Multiplying (7+2i)(7–2i);
Performing i5;
Performing i24;
Performing i35;
Dividing 4+2i/5-3i;
Dividing 5/2i;

Simplifying i8; i12; i16; i20
Simplifying –i42; i31

Simplifying i–42; i–28; – i–13

## Ch 8.2  Trigonometric (Polar) Form of Complex Numbers

Trigonometric Form z=r(cosθ+i sinθ) of Complex Numbers :

Plotting 2+3i; 4i; –3; –1–4i;
Graphing z=12[cos(π/6) +i sin(π/6)] and Converting it to Rectangular Form;
Graphing z=3csi(150°) and Converting it to Rectangular Form;
Writing z=–2+2i in Trigonometric/Polar Form
Writing z=3+5i in Trigonometric/Polar Form

Converting 2+2i into Trigonometry/Polar Form

Converting –4i into Trigonometry/Polar Form

Converting (3–i)2 into Trigonometry/Polar Form

Converting 3[cos(135°)+isin(135°)] to Rectangular Form
Converting 4[cos(270°)+i sin(270°) to Rectangular Form

Writing the Complex Number 2[cos(2π/3)+isin(2π/3)] in Rectangular Form

## Ch 8.3  The Product and Quotient Theorems

Multiplying {2[cos(π/3) +i sin(π/3)]}{3[cos(11π/6)+i sin(11π/6)]};

Dividing 24[cos(150°)+i sin(150°)]/6[cos(30°)+i sin(30°)]

Dividing 6[cos(60°)+i sin(60°)]/3[cos(90°)+i sin(90°)]
Multiplying {2[cos(45°)+i sin(45°)]}{5[cos(30°)+i sin(30°)]}

Dividing 3+3i / cos(90°)+i sin(90°)

## Ch 8.4  De Moivre’s Theorem; Powers and Roots of Complex Numbers

De Moivre’s Theorem;

Evaluating(-2√3-2i)²;

Definition of Roots of Complex Numbers;

Finding the First Fourth Roots of 16 in Rectangular Form a+bi and Trigonometric Form z=r[cos(θ)+i sin (θ)];

Finding the First Third Roots of 4√3-4i in a+bi form and Trigonometric Form (part 1)

Finding the First Third Roots of 4√3-4i in a+bi form and Trigonometric Form (part 2) ;

Rewriting in Rectangular Form;

Using De Moivre’s Theorem to Compute (1/2 – √3/2i)5

Using De Moivre’s Theorem to Compute (2+2i)4(√3+i)2

Determining all 4th Roots of z= –8+8i √3

## Ch 8.5  Polar Equations and Graphs

Plotting P(3, 30°), Q(2, 3π/2), R(–4, 60°), S(–1, –π/3 ) on the Polar Coordinate System;
Listing the Given Point in a 4 Different Way on the Polar Coordinate;
Listing A(–1, –1) in a 2 Different Way on the Polar Coordinate;
Writing E(–1,π/3 ) in Rectangular Coordinate

Listing the Given Point in a 4 Different Way on the Polar Coordinate in Degrees

Listing the Given Point in a 4 Different Way on the Polar Coordinate in Radians

Animation: Comparing Polar and Rectangular Coordinates

Converting (4, 1) and (–2, 3) to Polar Coordinates Using Degrees

Convert (–3, 3) and (–4, –3) to Polar Coordinates Using Radians

Finding the Rectangular and Polar Equation of a Circle from a Graph

Finding the Polar Equation for a Horizontal Line

Writing the Equation Line 3x–2y=6 in Polar Form

Graphing Polar Equations :
r=3;
θ=π/3;
r=3sin(θ);
r=3cos(θ);

r=4sin(3θ);
Showing Circles, Lemniscates, Limacons, and Rose Curves

Graphing Polar Equation y=3cos(2θ): Part 1 and 2

Graphing Polar Equation y=4sin(3θ) on the TI84 Graphing Calculator

Converting Polar Equations to Rectangular Equations and Graphing them :
r=tan(θ)sec(θ);
r=4cos(θ);
r=2/3sin(θ)-cos(θ)

Converting Rectangular Equation r sin2(θ)=2cos(θ) to Polar Equation and Graphing them

## Ch 8.6  Parametric Equations, Graphs, and Applications

Parametric Curves – Basic Graphing :
Graphing the Parametric Equation x=2t–1, y= –t+3, -4≤t≤5;
Graphing the Parametric Equation x=4cos(t), y=4sin(t), 0≤t≤2π

Steps of Converting Parametric Equations of Rectangular Equations;
Converting Parametric Equations x=4t+4, y=t+2 to Rectangular Equation, Stating the Domain and Graphing it;
Converting Parametric Equations x=√t, y=t-5 to Rectangular Equation, Stating the Domain and Graphing it;
Converting Parametric Equations x=e-t, y=e-t-1 to Rectangular Equation, Stating the Domain and Graphing it;
Converting Parametric Equations x=3cos(t), y=3sin(t) to Rectangular Equation and Graphing it;

Sketching the Curve Given by x=1–t , y=t2; –2≤t≤2
Sketching the Curve Given by x=2t–2, y= –t+3
Sketching the Curve Given by x=4–6t, y=3t
Sketching the Curve Given by x=2t, y=t2+1, –∞≤t≤∞