# Trigonometry

Trigonometry Videos

Chapters and section order are based on Trigonometry, 9th Edition, Lial, Hornsby, Schneider, Addison-Wesley

Ch 1  Trigonometric Functions

Ch 1.1  Angles

Angle Basics – Ray, Positive Angle, Negative Angle, Degree, Acute, Obtuse, Right, Straight, and Reflex angle, Complementary Angle, Supplementary Angle

Animation: Types of Angles – Acute, Obtuse, Right, Straight, and Reflex angle

Adjacent, Complementary, Supplementary, and Vertical Angles

Degrees, Minutes, and Seconds

Animation: Angles in Standard Position

Finding the Quadrant in Which an Angle Lies :

Examples :45º, 195º, -20º, –72º, 340º, 120º

Examples :380º, 1240º, –445º

Examples :25º, –255º, –580º

# Angles in Standard Position Quadrantal Angle, and Coterminal Angles :

Coterminal Angles of 135º, 1070º, –65º, 90º

Coterminal Angles of 45º, 135º, –255º, 90º

Determining if the pairs of angles are Coterminal Angles

Determining Positive and Negative Coterminal Angle of 68º

# Ch 1.2   Angle Relationships and Similar Triangles

Angle Relationships and Types of Triangles – Vertical Angle, Interior Angles, Equilateral Triangle, Isosceles Triangle, Scalene Triangle, Right Triangle, Acute Triangle, Obtuse Triangle, Sum of Measures of Angles

Similar Polygons – Properties of Similar Polygons, Solve the Similar Triangles

Congruent and Similar Triangles – Conditions for Similar Triangles, Solve for Similar Triangles

Sum of Interior Angles in a Triangle

Ch 1.3  Trigonometric Functions

Introduction to Six Trigonometric Functions Using Triangles – Sine, Cosine, Tangent, Secant, Cosecant, Cotangent

Introduction to Six Trigonometric Functions Using Angles (in a Unit Circle) – Sine, Cosine, Tangent, Secant, Cosecant, Cotangent

Six Trigonometric Functions of Quadrantal Angles (0º, 90º, 180º, 270º)

# Finding The Trigonometric Function value based on a Coordinate :

Finding the Trigonometric Function values of Cotangent and Tangent with Coordinate of (3, 4)

Finding the Trigonometric Function value of Secant with Coordinate of (5, 6)

Finding the Trigonometric Function values of Sine and Cosine with coordinate of (7, –2)

Finding the Six Trigonometric Function Values with Coordinate of (10, –6)

# Ch 1.4  Using the Definitions of the Trigonometric Functions

Finding Trigonometric Function Values Given One Trig Value in a Right Triangle
cosθ=

# Finding the Area of the Trianglehttp://www.youtube.com/watch?v=AK3GgzWSHJc&feature=relmfu

Finding the Missing Length and Cotangent Value

# Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities, and use identities to solve for angles :

Determining sec θ Given cos θ=
Determining cot θ Given tan θ=5
Determining cos θ Given sec θ=
Finding sin θ and tan θ Given that cos θ=  and sin θ>0
Finding sin θ and cos θ Given that tan θ=  and θ is in Quadrant III

Finding sin θ and tan θ Given that cos θ=  and θ is in Quadrant IV
Finding sec θ and sin θ Given that cos θ=  and θ is in Quadrant II
Finding Six Exact Trigonometric Function Values Given the Point (–2, –4) on the Terminal Side of θ

Determining the Quadrant of the Terminal Side of an Angle Given Trig Function Signs

Finding Trigonometric Values Given One Trigonometric Value :

Finding sec θ, csc θ, tan θ and cot θ from sin θ=  and cos θ=

Finding cos θ from tan θ=  and cos θ<0

Finding the Five Remaining Trigonometric Function Values from cos θ=  and θ is in Quadrant IV

Ch 2  Acute Angles and Right Triangles

Ch 2.1  Trigonometric Functions of Acute Angles

Cofunction Identities

30–60–90 and 45–45–90 Triangles and the Relationship of the Sides of the Right Triangles :

sin 30º,cos 30º, tan 30, sin 60º,cos 60º, tan 60º, sin 45º,cos 45º, tan 45º

Using the Ratio of the Sides of 30–60–90 Triangle or 45-45-90 Triangle to solve the Triangles

Examples: Solve a 30–60–90 Right Triangle

Examples: Solve a 45–45–90  Right Triangle

Ch 2.2  Trigonometric Functions of Non-Acute Angles

Examples: Determine the Reference Angle for a Given Angle :

θ=221º, θ=347º, θ= –125º

θ=460º, θ=165º, θ= –40º, θ= –283º

Determining Trigonometric Function Values Using Reference Angles and Reference Triangles :

Reference Angles of 120º, 210º, –45º, 270º

Trigonometric Function of sin (–150º), cos 960º, tan 180º, csc 225º, sec (–240º), cot 540º from a Unit Circle

Trigonometric Function Values of 90º, 150º, –60º, from a unit Circle

Trigonometric Function Values of –990º from a unit Circle

Ch 2.3  Finding Trigonometric Function Values Using a Calculator

Determining Trigonometric Function Values on the Calculator :
sin 30º, cos 45º, tan (–264º), sec (102.5º), csc (432º), cot (–23.45º)

Using Inverse Trigonometric functions to Find Angles :

Solving for θ for sin θ=0.7523, tan θ=3.54, and Find the Sides and Angles of a Right Triangle

Solving for θ for cos-1( ) and tan-1(–1)

# Ch 2.4  Solving Right Triangles

Example: Determining the Measure of an Angle of a Right Triangle Using a Trig Equation

Solving the sides and angles of Right Triangles

Applications : Angle of Elevation

Finding the Height of an Object

Finding the Height of a Tree

Finding the Height of a Flagpole (Part 1 of the video)

Simple Distance Problem – A Hiking Problem

Determining the Speed of a Boat

# Ch 2.5  Further Applications of Right Triangles

Applications : Finding the Distance of a Ship on an Angle of Bearing (Part 2 of the video)

Finding the Height of a Building

Finding the Length of x Using Right Triangle Trigonometry

# Ch 3  Radian Measure and Circular Functions

Showing the Relationship between Degree and Radian, and Converting Angles from Degree 120º to Radian

Examples: Converting Angles 135º. –60º, 15º, 48º to Radian

Examples: Converting Angles of , , and 2.1 in Radian to Degree

Examples: Determining Coterminal Angles of  and

Examples: Determine Six Trig Function Values of  Using Reference Triangles

# Ch 3.2  Applications of Radian Measure

Arc Length and Area of a Sector :

Finding Arc Length from r=9.5, θ=120º
Finding the Difference of Arc Length of d=12ft and d=11.81ft
Finding Area of a Sector from r=450, θ=240º

Finding Arc Length from r=3cm, θ=120º
Finding the Distance of Earth’s Path Around the Sun in One Month from  r=93 Million Miles

Examples: Area of a Sector and Area Bounded by a Chord and Arc
Finding Area of a Sector from r=8, θ=110º
Finding Area of a Sector from r=12, θ=80º

# Ch 3.3  The Unit circle and Circular Functions

A way to remember the Entire Unit Circle for Trigonometry

Tricks to remember Angles in Radians

# Ch 3.4  Linear and Angular Speed

Linear Velocity and Angular Velocity

Example: Determine the Number of Revolutions Per Second of a Car Tire

Example: Determine Angular and Linear Velocity of Two Particles Running in Concentric Circles

# Ch 4.1  Graphs of the Sine and Cosine Functions

Graphing the Sine and Cosine Function

Graphing Sine and Cosine with Different Coefficients (Amplitude and Period) :

Finding the Period of y=cos(3x), and y=sin( )
Finding and Graphing the Amplitude and Period of  y=sin( ), y= cos(2x) and y= –2sin(3x)

Finding the Amplitude and Period of y= –4cos(3x), y= sin(x) and y= cos( x)

Describing the Transformation of y= –3cos(6x)

Graphing y = –2 cos(2x)

# Ch 4.2  Translations of the Graphs of the Sine and Cosine Functions

Graphing Sine and Cosine with Phase Shifts (Horizontal Translation), Example 1 and Example 2 :

Phase Shifts (Horizontal Translation) for y=cos(x–4), y=sin[2(x+1)], and y=cos(10x+30)

Which Graph Most Closely Resembles the Graph of y= –2sin(x–π)

Horizontal and Vertical Translations of Sine Cosine :

Vertical Translation of y=sin (x)+1 and y=cos(x)–
Horizontal and Vertical Translation and Graphing of y=sin(x– )+1, and y= – +cos(x+ )

Horizontal and Vertical Translation and Graphing of y= +sin(x+ )

Graphing Sine and Cosine with Different Coefficients (Amplitude, period, and Vertical Translation) : y=2sin( )–1

Graphing Sine and Cosine with Different Coefficients (Amplitude, period, Horizontal Translation, and Vertical Translation) :

y=4cos(4x–8)–1

y=2sin(2 (x+ ))–1
y= cos( x– )+2
y=4–sin (π(x+1)

Example: Describe the Transformations of Cosine Function from a Graph : Fill Out Amplitude, Period, Horizontal Transformation (Phase Shift), Vertical Transformation (Vertical Shift), Midline

# Ch 4.3  Graphs of the Tangent of Cotangent Functions

Graphing the Tangent Function y=tan θ

Example: Graphing the Tangent Function y=tan θ
Using the Unit Circle and the Reciprocal Identity

Graphing the Cotangent Function y=cot θ

Graphing Tangent and over a Different Period y=tan(4x)

Graphing Tangent and Cotangent over Different Periods and Amplitude :

y=tan(3x)
y=cot( )
y=2tan( )
y= cot(2x)

y=cot(πx)
y=2cot(πx)

Example: Graphing a Transformation of the Cotangent Function y=2cot( x)

Identifying a Trigonometric a Function from Its Graph

# Ch 4.4  Graphs of the Secant and Cosecant Functions

Graphing y=csc θ Using y=sin θ
Graphing y=sec θ Using y=cos θ

Graphing y= –csc θ Using y=sin θ

Example: Graphing a Transformation of Cosecant Function
y=2csc(2πx+π)+3

Example: Determine the Domain of the Secant and Cosecant Functions Using the Unit Circle

Identifying a Trigonometric a Function from Its Graph

# Ch 5.1  Trigonometric Identities

Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities

Example: Verifying Pythagorean Identities for a Specific Angle :
Verifying sin2θ+cos2θ=1 for 150º
Verifying tan2θ+1=sec2θ for 45º

Negative Angle identities : sin(–θ), cos(–θ), tan(–θ), csc(–θ), sec(–θ), and cot(–θ)

Simplifying Trigonometric Expressions Using Identities :

sec(x)·cos(x)–cos2(x)

(csc2(x)–1)(sec2(x)·sin2(x))

sin2(x)·cos(x)·csc(x)

(1–cos2(x))(1+cot2(x))

Simplifying Trigonometric Expressions Involving Fractions

# Ch 5.2  Verifying Trigonometric Identities

Simplifying Products of Binomials Involving Trigonometric Functions :

(cos(x)–1)(cos(x)+1)
(sec(x)+tan(x))(sec(x)–tan(x))

[tan(θ)+cos2(θ)+sin2(θ)][tan(θ)–cos2(θ)–sin2(θ)]

Factoring Trigonometric Expressions :

sin2(θ)+cot2(θ) sin2(θ)

# Verifying that each Trigonometric Equation is An Identity :

cos(θ)tan(θ)=sin(θ);
;
;
; sin4(θ)–cos4(θ)=2sin2(θ)–1

tan(x)+cot (x)=sec(x)csc(x);
;

cos(θ)tan(θ)=sin(θ);
cos(x)[csc(x)tan(x)]=cot(x)+sin(x);
;

# Ch 5.3  Sum and Difference Identities for Cosine

Sum and Difference Identities for Cosine :

Finding cos(A+B) if A=  in Quadrant II and B=  in Quadrant I
Determining the Exact Value of cos15°
Determining the Exact Value of cos
Determining the Exact Value of cos40°cos50°–sin40°sin50°

Verifying Sum Identity for Cosine (2nd Example in the Video) :

cos(x+ )= (cos(x) – sin(x))

Finding the function values and the Quadrant of A–B :

Finding cos(x–y) from tan(x)=  and cos(y)=  Where x and y Are in Quadrant IV

Cofunction Identities :

sin18º; tan65º, csc84º
cos( ); cot( ); sec( )

Cofunction Identities : Solving Trigonometric Equations :

cos (2θ+16°)=sin(θ+11°)
cot(θ)=tan(θ+ )

Ch 5.4  Sum and Difference Identities for Sine and Tangent

Sum and Difference Identities for Sine :

Determining the Exact Value of sin105°
Determining the Exact Value of sin( )

Determining the Exact Value of sin75°

Determining the Exact Value of sin20°cos40°–cos20°sin40°-

Sum and Difference Identities for Tangent  :

Determining the Exact Value of tan(–105°)
Determining the Exact Value of tan( )
Using an Identity to Write tan(π–θ) as a Single Function of θ

Determining the Exact Value of tan( )

Sum and Difference Identities to Simplify an Expression :

Simplify tan(x+4π)

Simplify tan(4π–x)

Finding the function values and the Quadrant of A+B :

Finding tan(2x) if sin(x)=  and x is in Quadrant I

Finding sin(2x) if tan(x)=  and x is in Quadrant I

Ch 5.5  Double-Angle Identities

Double Angle Identities :

Finding the Exact Value of cos(2A), sin(2A) and tan(2A) and Quadrant of 2A  if  sin(A)=  is in Quadrant II
Finding cos(A) given cos(2A)=  where 2A is in Quadrant III

Using Double Angle Identities to Simplify and then Evaluate :
cos2( )–sin2( )
2sin( )–cos( )
2cos2( )–1

Example : Determining Double Angle Trigonometric Function Values with Given Quadrant :

Finding sin(2θ), sin(2θ) and tan(2θ) from cos( ) and θ is in Quadrant II

Finding sin(2θ), sin(2θ) and tan(2θ) from tanθ=( ) and sinθ<0

Verifying Double Angle Identity (1st Example in the Video) :

(sinA+cosA)2= sin(2A)+1

Example: Using Double Angle Identity :

Product to Sum and Sum to Product Identities :

Product to Sum: sin(–4θ) sin(8θ)
Product to Sum : 2cos(  )–cos(  )
Product to Sum : sin(  )–cos(  )
Sum to Product :cos(9x)+cos(4x)
Sum to Product : sin( )–sin( )

Ch 5.6   Half-Angle Identities

Half Angle Identities  :

Finding the Exact Value of sin(  )
Determining the Exact Value of cos105°
Finding cos( ), sin( ) and tan( ) from cosA=( ) in Quadrant II

Example: Rewriting a Trig Expression Using a Half Angle Identity :

(sin(5x))2
(cos(2x))4

Example: Determine a Cosine Function Value Using a Half Angle Identity :

Example: Determining a Sine Function Value Using a Half Angle Identity :

sin(112.5°)

Example: Finding a Sine Function Value from a Cosine Function Value Using a Half Angle Identity :
Finding sin( ) if cos(a)=( ) for 0°≤a≤90°

Example: Determining a Tangent Function Value Using a Half Angle Identity :

Verifying Half Angle and Double Angle Identities for Sine (3rd Example in the Video) :

Ch 6  Inverse Circular Functions and Trigonometric Equations

Ch 6.1  Inverse Circular Functions

Inverse Functions

Animation: Illustrate why a function must be one-to-one to have an inverse function

Introduction to Inverse Sine, Inverse Cosine, and Inverse Tangent

Introduction to Inverse Cosecant, Inverse Secant, and Inverse Cotangent :

The Domain of y=sin(x) is [ , ] and the Range is [–1, 1]
The Domain of y=sin-1(x) is [–1, 1] and the Range is [ , ]
The Domain of y=cos(x) is [0, π] and the Range is [–1, 1]
The Domain of y=cos-1(x) is [–1, 1] and the Range is [0, π]
The Domain of y=tan(x) is [ , ] and the Range is [–∞ ∞]
The Domain of y=tan-1(x) is [–∞, ∞] and the Range is [ , ]
Finding the Exact Value of y=arcsin( )
Finding the Exact Value of y=arccos( )
Finding the Exact Value of y=tan-1(–1)

Finding the Exact Inverse Function Values Involving Inverse Sine, Inverse Cosine, and Inverse Tangent :

Evaluating y=sin-1( )
Evaluating y=cos-1(0)
Evaluating y=arctan(–1)

Evaluating y=sin-1(sin( ))
Evaluating y=sin-1(cos( ))
Evaluating y=tan-1(sin( ))

Finding the Exact Inverse Function Values Involving Inverse Cosecant, Inverse Secant, and Inverse Cotangent  :

Evaluating csc-1( ) in Degree and Radian

Evaluating arcsec(–2) in Degree and Radian

Evaluating sin-1( )
Evaluating sec-1(2)
Evaluating csc-1( )
Evaluating cot-1(–1)

Determining the Exact Value without a Calculator (Part 1 of the video) :
arcsec(2)
arccsc( )
arccot(–1)

Finding a Inverse Cotangent Value in Degrees and Radians Using a Calculator :

arccot(–3.6)

Finding the Exact Function Values Involving Inverse Sine, Inverse Cosine, and Inverse Tangent :

sin(arccos( ))
cos(arctan( ))
sin(sin-1( )+tan-1(√–3))
Finding the Angle the Ladder Makes with the Ground
Finding the Maximum Angle of Elevation to Maximize a Shot Putter Distance

Evaluating sin(sin-1( ))
Evaluating cos(cos-1( ))
Evaluating tan(cos-1( ))

Evaluating tan(sin-1( ))

Evaluating sin(tan-1(–7))

Evaluating sin(tan -1( )) and Assume u>0

Finding an Exact Sine Function Value Containing an Inverse Cosine – Double Angle :
sin(2arccos( ))

Finding the Exact Function Values Involving Inverse Cosecant, Inverse Secant, and Inverse Cotangent  Without a Calculator (Part 2 of the video) :
csc(arccot(u))
cos(sec-1( ))
sec(arccot( ))

Ch 6.2  Trigonometric Equations I

Solving a Trigonometric Equation by Linear Method :

Solving Each Equation on the Interval [0, 2π) and then Over All Radian Solutions
2sinθ–1=0
2cosθ+√2=0
√3tanθ–1=0
4cosθ–6=cosθ

Solving sin(x)+ =0 on the Interval [0, 360°)

Solving 2cos(x)sin(x)=sin(x) on the Interval [0, 360°)

Solving 3tan2(x)–1=0 on the Interval [0, 2π)

Solving 3=20sin(x–3)+1 on the Interval [0, 2π) (Part 1 of the video)

Solving  on the Interval [0, 2π) (Part 1 of the video)

Example: Solving Trigonometric Equation: sin(x)=cos(x)

Solving a Trigonometric Equation by Factoring :

Solving tan2θ–1=0 on the Interval [0, 2π)
Solving 2cos2θ–√3cosθ=0 on the Interval [0, 2π)
Solving 2sin2θ=–3sinθ–1 on the Interval [0, 2π)

Solving 2cos2(x)–sin(x)=1 on the Interval [0, 2π)

Solving cos2(x)–cos(x)–2=0 on the Interval [0, 2π)

Solving a Trigonometric Equation by Trigonometric Identities :

Solving cos2(x)–sin2(x)=  on the Interval [0, 360°)
Solving tanθ+√3=secθ on the Interval [0, 2π)

Solving sec2(x)–2tan(x)=4 on the Interval [0, 2π) (Part 2 of the video)

Solving a Trigonometric Equation Using the Calculator :

Solving sin(x)–0.32=0 on the Interval [0, 2π)

Solving cos(x)+0.85=0 on the Interval [0, 2π)

Solving Applications Problems :

The Function T(x)=19sin( x – )+53  Modeling the Average Monthly Temperature of Water in a Mountain Stream
The Function S(x)=1600cos( x+ )+5100 Modeling the Average Monthly Sales in the Month x

Determining the Height of an Object Using a Trigonometric Equation

Ch 6.3  Trigonometric Equations II

Solving a Trigonometric Equation by Double Angle :

Solving cos(2x)+sin2(x)–3cos(x)=1 on the Interval [0, 360°) (Part 3 of the video)

Solving cos(2θ)–cos(θ)=0 on the Interval [0, 2π)
Solving sin(θ)–sin(2θ)–=0 on the Interval [0, 2π)
http://www.youtube.com/watch?v=8FRly0POPD8 (Part 3 and 4 of the video)

Solving sin(2x)=cos(2x)+1 on the Interval [0, 2π) (Part 3 of the video)

Solving cos(2x)=cos(x) on the Interval [0, 2π)

Solving sin(2x)=2cos2(x) on the Interval [0, 2π)

Solving a Trigonometric Equation by Half Angle (Part 1 of the video):
sin( )=√2–sin( )

Solving a Multiple-Angle–Trigonometric Equation by Single Angle :

Solving 4cos(4x)=2 on the Interval [0, 2π)

Solving 5sin(3x)=2 on the Interval [0, 2π)

Solving 2cos(3x)–√3=0 on the Interval [0, 360°) (Part 2 of the video)

Ch 6.4  Equations Involving Inverse Trigonometric Functions

Solving the Equation for Secant x :
√5+2sec(3x)=y

Solving a Trigonometric Equation with an Inverse Trig Function :

4arctan(x)=π
cos-1(x)=sin-1( )
http://www.youtube.com/watch?v=GqVopraSumU  (Part 2 and 3 of the video)

cos-1( )=( )

cos-1(x)=sin-1( )

Ch 7  Applications of Trigonometry and Vectors

Ch 7.1  Oblique Triangles and the Law of Sines :

The Law of Sines: The Basics :
Solving the Triangle with A=48°, B=54°, a=12.5feet

Solving the Triangle with C=102°, b=5m, c=18m

The Law of Sines: Applications :

Finding the Distance Across the Canyon

Finding the Distance of the Campsite from the Base of the Mountain and the Time Needed to the Base

Determining the Length of x

Determining the Distance from the Tower to the Airplane
Determining the Elevation of the Airplane

The Area of a Triangle using Sine :

Determining the Area of a Triangle with A=35°, B=82°, a=6cm, b=15cm
Determining the Area of a Triangle with B=72°, a=23.7ft, b=25.2ft

Ch 7.2  The Ambiguous Case of the Law of Sines

The Law of Sines: The Ambiguous Case :

Solving the Triangle with C=82°, a=11m, c=7m
Solving the Triangle with A=88°, a=110ft, c=54ft
Solving the Triangle with B=40°, b=22cm, a=30cm

Ch 7.3  The Law of Cosines

The Law of Cosines :

Solving Triangle ABC with B=73.1°, a=24.2ft, c=43.7ft
Solving Triangle ABC with a=25.4cm, b=42.8cm, c=59.3cm

The Law of Cosines: Applications – Determining a Side from Two Sides and an Angle :

Determining the Diagonal of a Parallelogram

Determining the Length of the Tunnel and the Bid Amount after 20% profit off the Cost
Determining the Duration of Service a Person gets when She is Driving Past a Transmission Tower at a Certain Speed

Determining the Flying Distance of A plane From Her Starting Position

The Law of Cosines: Applications – Determining the Largest Angle of the Three Sides

Heron’s Area Formula :

Determining the Areas of Triangles Using Heron’s Area Formula

Ch 7.4  Vectors, Operations, and the Dot Product

Introduction to Vectors :

Vector Basics – Drawing Vectors/ Vector Addition

Finding the Components of a Vector :

Finding x and y Components of a Vector with a Magnitude of 8 and at an Angle of 60° from the Origin

Finding x and y Components of Vectors

Vector Basics – Algebraic Representations :

Drawing the Vector =<2,3>
Drawing the Vector   =<–3,4>
Finding the Vector  of (2, 1) and (5, 6)

Adding + +  where =<2, 2>, =<3, 0> and =<0, –6>
Multiplying 10  where =<1, 4>
Simplifying –2  where  =<1,4> and =<–2, 3>

Magnitude and Direction of a Vector :

Finding the Magnitude and Direction of Vector <–3, 4>

Finding the Magnitude and Direction of Vector <–2, –5>

Finding the Magnitude and Direction of Vector <2, 6> and <3, –10>

The Dot Product :

Finding ·  where =<2, 5> and =<–3, 1>
Finding the Angle of =6i–2j–3k and =i+j+k
Determining if =<2, 4> and =<4, –2> are Orthogonal

Ch 7.5  Applications of Vectors

Vectors: Applications :

Finding the Distance between the Ship and the Port and its Bearing
Finding the Direction and Magnitude of the Resultant Force of Two Trucking Pulling on a Truck Stuck in the Mud

Ch 8  Complex Numbers, Polar Equations, and Parametric Equations

Ch 8.1  Complex Numbers

Introduction to Complex Numbers :

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

Complex Number Operations :

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

Rewriting Powers of ‘i’ :

Simplifying i8; i12; i16; i20
Simplifying –i42; i31; i–42; i–28; – i–13

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 +i sin ) and Converting it to Rectangular Form
Graphing z=3csi150° and Converting it to Rectangular Form
Writing z=–2+2i in Trigonometric Form
Writing z=3+5i in Trigonometric Form

Converting from Rectangular Form to Trigonometric Form z=x+yi :

Converting 2+2i into Polar Form

Converting –4i into Polar Form

Converting (3–i)2 into Polar Form

Converting from Trigonometric Form to Rectangular Form :

Converting 3(cos135°+i sin135°) to Complex Form
Converting 4(cos270°+i sin270°) to Complex Form

Ch 8.3  The Product and Quotient Theorems

The Product and Quotient of Complex Numbers in Trigonometric Form :

Multiplying [2(cos +i sin )][3(cos +i sin )]
Dividing

Dividing
Multiplying [2(cos45°+i sin45°)][5(cos30°+i sin30°)]

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

De Moivre’s Theorem : Raising a Complex Number to a Power :

Rewriting [2(cos +i sin )]3 in Rectangular Form
Using De Moivre’s Theorem to Compute ( – i)5

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

Determining the nth Roots of a Complex Number :

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

Ch 8.5  Polar Equations and Graphs

Introduction to Polar Coordinates :

Plotting P(3, 30°), Q(2, ), R(–4, 60°), S(–1, – ) 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, ) 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=
r=3sinθ
r=3cosθ

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=tanθsecθ
r=4cosθ
r=

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 :

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≤∞

Sketching the Curve Given by x=1+√t, y=t2–4t, 0≤t≤5
Sketching the Curve Given by x=√t, y=1–t