Chapter 8 Trigonometry

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
http://www.youtube.com/watch?v=NeTRNpBI17I

Adding (3+4i)+(7–5i);
Adding (–2+7i)–(–4+7i);
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;
http://www.youtube.com/watch?v=htiloYIILqs

Simplifying i8; i12; i16; i20
Simplifying –i42; i31
http://www.youtube.com/watch?v=VTAsbx5wBaI&feature=player_embedded

Simplifying i–42; i–28; – i–13
http://www.youtube.com/watch?feature=player_embedded&v=tbWfxuV-IeE

 

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
http://www.youtube.com/watch?v=Zha7ZF8aVhU

Converting 2+2i into Trigonometry/Polar Form
http://www.youtube.com/watch?feature=player_embedded&v=6z6fzPXUbSQ

Converting –4i into Trigonometry/Polar Form
http://www.youtube.com/watch?v=tAIxdEVuTZ8&feature=player_embedded

Converting (3–i)2 into Trigonometry/Polar Form
http://www.youtube.com/watch?v=XIYDO_weAVA&feature=player_embedded

Converting 3[cos(135°)+isin(135°)] to Rectangular Form
Converting 4[cos(270°)+i sin(270°) to Rectangular Form
http://www.youtube.com/watch?v=tl56MgKA3Z8&feature=player_embedded

Writing the Complex Number 2[cos(2π/3)+isin(2π/3)] in Rectangular Form
http://www.youtube.com/watch?v=9rxaasy96dw&feature=plcp

 

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°)]
http://www.youtube.com/watch?v=JNiLUgJcluU

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°)]}
http://www.youtube.com/watch?v=in_dAlpjUKk&feature=player_embedded

Dividing 3+3i / cos(90°)+i sin(90°)
http://www.youtube.com/watch?feature=player_embedded&v=QxboPshf__g

 

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)
http:www.youtube.com/watch?v=YLoSvy673L8&list=PL554F9F460401DF50&index=36&feature=plpp_video

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

Solving x²+9i=0
http:www.youtube.com/watch?v=YLoSvy673L8&list=PL554F9F460401DF50&index=36&feature=plpp_video

Rewriting in Rectangular Form;

Using De Moivre’s Theorem to Compute (1/2 – √3/2i)5
http://www.youtube.com/watch?v=Sf9gEzcVZkU

Using De Moivre’s Theorem to Compute (2+2i)4(√3+i)2
http://www.youtube.com/watch?v=irwItP7PiQg&feature=player_embedded

Determining all 4th Roots of z= –8+8i √3
http://www.youtube.com/watch?v=0Gyv9ce7f8I

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
http://www.youtube.com/watch?v=-tZR3ggdoIU

Listing the Given Point in a 4 Different Way on the Polar Coordinate in Degrees
http://www.youtube.com/watch?v=ds-YLa7Yd0k

Listing the Given Point in a 4 Different Way on the Polar Coordinate in Radians
http://www.youtube.com/watch?v=0xh32SzuUSU

Animation: Comparing Polar and Rectangular Coordinates
http://www.youtube.com/watch?v=A-AEuohaP4o

Converting (4, 1) and (–2, 3) to Polar Coordinates Using Degrees
http://www.youtube.com/watch?v=Vg88CWOlDTY

Convert (–3, 3) and (–4, –3) to Polar Coordinates Using Radians
http://www.youtube.com/watch?v=_rSC6Im1u1U

Finding the Rectangular and Polar Equation of a Circle from a Graph
http://www.youtube.com/watch?v=ZFNSMfWvVNY&feature=youtu.be

Finding the Polar Equation for a Horizontal Line
http://www.youtube.com/watch?v=_Ku4ZP4bmps&feature=youtu.be

Writing the Equation Line 3x–2y=6 in Polar Form
http://www.youtube.com/watch?v=xrsJNoq4oN4&feature=youtu.be

Graphing Polar Equations :
r=3;
θ=π/3;
r=3sin(θ);
r=3cos(θ);
http://www.youtube.com/watch?v=X6M7yfH6w_Y

r=4sin(3θ);
Showing Circles, Lemniscates, Limacons, and Rose Curves
http://www.youtube.com/watch?v=mTxxNQWWlBc

Graphing Polar Equation y=3cos(2θ): Part 1 and 2
http://www.youtube.com/watch?v=mDT_DG_A0JA&feature=player_embedded
http://www.youtube.com/watch?v=GMcRqtm4mNo&feature=player_embedded

Graphing Polar Equation y=4sin(3θ) on the TI84 Graphing Calculator
http://www.youtube.com/watch?v=PZwiiZQhM0c&feature=youtu.be

Converting Polar Equations to Rectangular Equations and Graphing them :
r=tan(θ)sec(θ);
r=4cos(θ);
r=2/3sin(θ)-cos(θ)
http://www.youtube.com/watch?v=IKbRiU7kL2w

Converting Rectangular Equation r sin2(θ)=2cos(θ) to Polar Equation and Graphing them
http://www.youtube.com/watch?v=A1y81g78VoM&feature=youtu.be

 

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
http://www.youtube.com/watch?v=Fz6p4aC9e2Q

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;
http://www.youtube.com/watch?v=tW6N7DFTvrM

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≤∞
http://www.youtube.com/watch?v=VqsGKIWsT20

Sketching the Curve Given by x=1+√t, y=t2–4t, 0≤t≤5
Sketching the Curve Given by x=√t, y=1–t
http://www.youtube.com/watch?feature=player_embedded&v=tsnHL1Lb5MU

Graphing Parametric Equations on the Calculator TI84

x=4-t, y=2t;
x=√t; y=t-5, t≥0;
x=5sin(t), y=2cos(t), 0≤t≤2π;
x=6cos³(t), y=6sin³(t), 0≤t≤2π;
y=4cos(t), y=2sin(4t), 0≤t≤2π;
http://www.youtube.com/watch?v=4Y14XhPD7Os