High School Mathematics: PreCalculus

Video Lectures

Displaying all 145 video lectures.
Lecture 1
Function vs Relation
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Function vs Relation
I introduce the concepts of Function and Domain. I compare and contrast them with coordinates, graphs, and equations through many examples.
EXAMPLES at 5:14 11:02 16:05 18:12 21:33 23:05
Lecture 2
Visually identifying key characteristics of graphs
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Visually identifying key characteristics of graphs
How to identify key features of graphs such as relative maximum and minimum, zeros, domain, range, and y-intercept. We will also state the intervals where the graph is increasing, decreasing, and constant.
Lecture 3
Even Odd Polynomial Functions & Symmetry
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Even Odd Polynomial Functions & Symmetry
Definining Even and Odd functions
Newer video with many more examples at https://www.youtube.com/watch?v=WJvwUaRQdiY
Lecture 4
Even and Odd Functions Many Examples
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Even and Odd Functions Many Examples
I explain how to determine if a function is even, odd, or neither and work through examples at
Even Functions 3:39 5:03 8:33 10:42 11:23
Odd and Even Functions, or Neither 15:27 17:36 19:57 22:47 29:24
Please excuse the static from my mic. I have made adjustments to avoid this in the future.
Lecture 5
Piecewise Functions
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Piecewise Functions
Introduction to piecewise functions, graphing, domain, and range.
Lecture 6
Difference Quotient
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Difference Quotient
PreCalculus introduction to the Definition of the Difference Quotient
Lecture 7
Rate of Change Slope & Point Slope Equation of Lines
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Rate of Change Slope & Point Slope Equation of Lines
Introduction to finding slope and graphing lines.

Please check out my other lessons about Linear Functions:
Graphing Lines in Slope-Intercept form y=mx+b
https://www.youtube.com/watch?v=xyVJZKu7Euw
Graphing lines using X & Y Intercepts
https://www.youtube.com/watch?v=KtgddI4qIok
Equations of Lines and Graphing
https://www.youtube.com/watch?v=Wmi1EysOHUQ
Equations of parallel and perpendicular lines
https://www.youtube.com/watch?v=ECljEiIgX_M
Regression Lines and Correlation with TI-84
https://www.youtube.com/watch?v=L_h7nqNqgPs
Linear Functions, their Attributes, & Interpreting Slope and y-Intercept
https://www.youtube.com/watch?v=qbNTy0oMq1c

Check out www.ProfRobBob.com, there you will find my lessons organized by class/subject and then by topics within each class.
Lecture 8
Equations of Lines and Graphing
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Equations of Lines and Graphing
Continued discussion about equations of lines and the three equation forms. Graphing of linear functions also included.
Please check out my other lessons about Linear Functions:
Graphing Lines in Slope-Intercept form y=mx+b
https://www.youtube.com/watch?v=xyVJZKu7Euw
Graphing lines using X & Y Intercepts
https://www.youtube.com/watch?v=KtgddI4qIok
Slope and Equations of Lines
https://www.youtube.com/watch?v=xf8-BEdoSss
Equations of parallel and perpendicular lines
https://www.youtube.com/watch?v=ECljEiIgX_M
Regression Lines and Correlation with TI-84
https://www.youtube.com/watch?v=L_h7nqNqgPs
Linear Functions, their Attributes, & Interpreting Slope and y-Intercept
https://www.youtube.com/watch?v=qbNTy0oMq1c

Check out www.ProfRobBob.com, there you will find my lessons organized by class/subject and then by topics within each class.
Lecture 9
Equations of Parallel and Perpendicular Lines
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Equations of Parallel and Perpendicular Lines
An introduction to finding the equations of lines that pass through a point and are parallel or perpendicular to a given line.
Please check out my other lessons about Linear Functions:
Graphing Lines in Slope-Intercept form y=mx+b
https://www.youtube.com/watch?v=xyVJZKu7Euw
Graphing lines using X & Y Intercepts
https://www.youtube.com/watch?v=KtgddI4qIok
Slope and Equations of Lines
https://www.youtube.com/watch?v=xf8-BEdoSss
Equations of Lines and Graphing
https://www.youtube.com/watch?v=Wmi1EysOHUQ
Regression Lines and Correlation with TI-84
https://www.youtube.com/watch?v=L_h7nqNqgPs
Linear Functions, their Attributes, & Interpreting Slope and y-Intercept
https://www.youtube.com/watch?v=qbNTy0oMq1c

Check out www.ProfRobBob.com, there you will find my lessons organized by class/subject and then by topics within each class.
Lecture 10
Average Rate of Change
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Average Rate of Change
Introduction to the average rate of change formula.
Lecture 11
Transformation of Functions
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Transformation of Functions
I cover transformations of vertical and horizontal slides, vertical stretch, horizontal and vertical reflections to help you draw quick sketches of some common graphs.
Lecture 12
Composition of Functions
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Composition of Functions
I introduce composition of functions and discuss domain.
Lecture 13
Determining Domain of Functions
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Determining Domain of Functions
I discuss finding domain of functions with the use of graphs or graphing calculators. The domain of the last example is no solution.
Identifying Domain & Range from a Graph
https://www.youtube.com/watch?v=1UAKQ_2yfxI&list=PLGbL7EvScm...
Lecture 14
Combining Functions & Function Operations
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Combining Functions & Function Operations
I go over combining functions through addition, subtraction, multiplication, and division. I also explain the restriction on domain of these combined functions
Lecture 15
Inverse Functions
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Inverse Functions
A graphic and algebraic approach to finding inverse functions.
Lecture 16
Modeling with Functions Part 1
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Modeling with Functions Part 1
We model real life scenarios of sales and volume of a box with functions. These type of PreCalculus questions will help to prepare for the sections of Calculus called Related Rates.
Lecture 17
Modeling with Functions Part 2
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Modeling with Functions Part 2
I continue with two more examples of Modeling with Functions which relate the area of regions with their perimeter. These type of questions will help to master the type of word problems you may see in the Related Rates sections of your Calculus book.
Lecture 18
Modeling with Functions Part 3
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Modeling with Functions Part 3
I do two more examples: first involving relating surface area to the volume of a rectangular solid , and second looking at the volume of a cone in terms of it's radius and then it's hight. These type of PreCalculus questions will help to prepare for the sections of Calculus called Related Rates.
Lecture 19
Complex (imaginary) Numbers Part 1
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Complex (imaginary) Numbers Part 1
I introduce complex numbers, show how i-squared is -1, show how to graph them, and then go through a number of algebraic examples showing how to work with them.
Lecture 20
Complex (imaginary) Numbers Part 2
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Complex (imaginary) Numbers Part 2
I restart and finish the example from Complex Numbers Part 1. I solve a quadratic equation and find a solution that is a pair of complex numbers. I then graph the parabola to explain why we found the solution to be imaginary.
Lecture 21
Graphing Parabolas w/ vertex & intercepts
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Graphing Parabolas w/ vertex & intercepts
I introduce the standard form of parabola and relate it to the transformations we just learned. I then graph a parabola in it's general form finding its vertex and X & Y intercepts.
Lecture 22
Parabola Applications Maximizing Minimizing Reflectors, etc
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Parabola Applications Maximizing Minimizing Reflectors, etc
I give examples of how parabolas apply to real life and work through two example...one about maximum height of a thrown object, and a second about maximizing area of a rectangle with a fixed perimeter
Lecture 23
Polynomial Graphs Part 1
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Polynomial Graphs Part 1
I introduce polynomial functions and give examples of what their graphs may look like. I then go over how to determine the End Behavior of these graphs. Part 2 will include finding y-intercepts, x-intercepts and their multiplicity, determining even or odd, and sketching the function.

At minute 2:34 I say that f(x)=4 has an exponent of 0 instead of a degree of 0.

At 7:04 I incorrectly draw y=-1/x instead of y=1/x...though my explaining about what a polynomial graph is still correct otherwise. An annotation was added but you won't see it with an iPad, iPod, or iPhone.
Lecture 24
Polynomial Graphs Part 2
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Polynomial Graphs Part 2
I continue my introduction of graphing polynomials without the assistance of a graphing calculator. I go over x-int & y-int, and I remind you how to determine if a function is even or odd to aid in graphing.
Lecture 25
Basic graphing with a TI-NSPIRE
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Basic graphing with a TI-NSPIRE
I show how to graph a parabola, find it's vertex, and x-intercept on a TI-NSPIRE.
Lecture 26
Factoring Overview
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Factoring Overview
I go over some common method of factoring binomials and trinomials.
Here you can find a selection of my Factoring Lessons
http://www.profrobbob.com/algebra-2/factoring-techniques
Lecture 27
Factoring by Grouping
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Factoring by Grouping
I work through 4 examples of factoring expressions and solving equations using factoring by grouping.
Here you can find a selection of my Factoring Lessons
http://www.profrobbob.com/algebra-2/factoring-techniques
Lecture 28
Solving Equations with 2 Absolute Value Functions or More
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Solving Equations with 2 Absolute Value Functions or More
I explain how to solve equations with two absolute value functions or more and then work through 3 examples at 3:48 14:13 22:57

In the last example after solving the equation I explain how to finish the problem if it were originally an inequality.
Lecture 29
Synthetic Division & Remainder Theorem
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Synthetic Division & Remainder Theorem
I introduce Synthetic Division and the Remainder Theorem. Synthetic division is a short cut to long division when you are dividing by a binomial in the form of (x-c). The Remainder Theorm is how you can use Synthetic Division to aid in evaluating polynomials at particular x values.
Lecture 30
Long Division of Polynomials
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Long Division of Polynomials
I introduce long division and go over two examples. I also preview how long division will allow you to solve higher order equations.
Lecture 31
Solving Higher Order Polynomials Pt 1 Rational Zeros Descartes Rule
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Solving Higher Order Polynomials Pt 1 Rational Zeros Descartes Rule
In a 2 part video I show you how to solve equations with a degree larger than 2 without the aid of a graphing calculator or computer.
Lecture 32
Solving Higher Order Polynomials Pt 2 Rational Zeros Descartes Rule
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Solving Higher Order Polynomials Pt 2 Rational Zeros Descartes Rule
In a 2 part video I show you how to solve equations with a degree larger than 2 without the aid of a graphing calculator or computer.
Lecture 33
Finding polynomials using the Linear Factorization Theorem
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Finding polynomials using the Linear Factorization Theorem
Given the degree, the real and imaginary solutions, and a point the graph goes through, I use the Linear Factorization Theorem to find the original polynomial function.



In the last problem, the question would have to specify that the coefficients of the polynomial are real. Otherwise complex solutions don't necessarily have to occur in conjugate pairs.
Lecture 34
Horizontal Asymptotes of rational equations
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Horizontal Asymptotes of rational equations
Using limits, I explain how to find how to find horizontal asymptotes of rational equations.
Lecture 35
Graphing Rational Functions with Slant Asymptotes
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Graphing Rational Functions with Slant Asymptotes
I show you how to find slant asymptotes of rational equations.
Lecture 36
Finding vertical asymptotes and holes of rational equations
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Finding vertical asymptotes and holes of rational equations
Finding vertical asymptotes and holes of rational equations.
Lecture 37
Graphing Rational Functions Part 1
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Graphing Rational Functions Part 1
I show how to graph a rational function with a vertical and horizontal asymptote, x-intercepts, y-intercepts, and even or odd symmetry.
Lecture 38
Graphing Rational Functions Part 2
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Graphing Rational Functions Part 2
This rational function I graph has a horizontal asymptoto, a vertical asymptote, and a hole.
Lecture 39
Solving Polynomial Inequalities
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Solving Polynomial Inequalities
I introduce solving polynomial inequalities.
Lecture 40
Solving Rational Inequalities
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Solving Rational Inequalities
I introduce solving rational inequalities. After showing you the steps you should follow for these types of problems I work through two examples. In the second example I write less than 0 in the second through forth line of work. That should read less than or equal to, an annotation correction has been made but you will not see this without FLASH.
Lecture 41
Graphing Exponential Functions w/ t-table or Transformations
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Graphing Exponential Functions w/ t-table or Transformations
I explain how to graph exponential functions using tables and transformations from a parent function. Properties of these graphs are also discussed such as domain, range, asymptotes, etc.
Lecture 42
Logarithm Introduction
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Logarithm Introduction
I introduce logarithms and show how to convert from logarithm form to exponential form. Yes, I know I mispelled logarithm on the chalk board. I blame it on the cold medicine I was taking!
Lecture 43
Change of Base Formula Logarithms
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Change of Base Formula Logarithms
I start with examples of evaluating log expressions that don't require a calculator, then compare that with a problem that does require the change of base formula.
Lecture 44
Using Properties of Logarithms to Expand Logs
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Using Properties of Logarithms to Expand Logs
I introduce the 3 properties of logarithms and use them in multiple examples about how to expand logarithmic expressions. Part 2 will include examples of condensing log expressions.

There is an annotation correction at 9:43 but you will not see it without Flash. I should have wrote -3 instead of -5.
Lecture 45
Using Properties of Logarithms to Condense Logs
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Using Properties of Logarithms to Condense Logs
In this video I continue covering the uses of the properties of logarithms to condense logarithmic expressions.
Lecture 46
Graphing Logarithms
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Graphing Logarithms
I introduce graphing Logarithms (which I know is mispelled!) and expain how they are an inverse of exponential functions. At 1:48 I misspoke and stated that all exponential growth functions pass through (1,0), but the point is actually (0,1)
Lecture 47
Solving Equations with Logarithms Pt 1
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Solving Equations with Logarithms Pt 1
PLEASE NOTE my final answer at 10:26 should be 3.968. There is an annotation correction but you can only see this if using FLASH.
In a three part video I do many examples of solving equations using logarithms to move the variable out of the exponent. In part two we remove logarithms from the equation to solve for X.
Lecture 48
Solving Equations with Logarithms Pt 2
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Solving Equations with Logarithms Pt 2
In a three part video I do many examples of solving equations using logarithms to move the variable out of the exponent. In part two we remove logarithms from the equation to solve for X.
Lecture 49
Solving Equations with Logarithms Pt 3
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Solving Equations with Logarithms Pt 3
In a three part video I do many examples of solving equations using logarithms to move the variable out of the exponent. In part three I do examples with the log function on both sides of the equation.
Lecture 50
Solving Compound Interest Problems
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Solving Compound Interest Problems
I go over examples involving compound growth and continuous compound growth. Some examples require the use of logarithms to solve for an unknown time period. I apologize that this video has some inconsistent rounding errors.
The answer to the first equation without round off error is $40387.39
Lecture 51
Standard Position Angles & Radians Part 1
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Standard Position Angles & Radians Part 1
Part 1 of 2. I introduce Standard Position Angles, Define Coterminal Angles, Quadrantal Angles, and the Radian Measure.
Standard Position Angles & Radians Part 2
https://www.youtube.com/watch?v=d0mmCN6rOXM&list=PL085526F86...
Lecture 52
Standard Position Angles & Radians Part 2
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Standard Position Angles & Radians Part 2
In part 2 I show how to convert angles from degrees to radians and radians to degrees. I then introduce the sixteen angles around what will be developed into the unit circle. On the angle of 330 degrees, I said 330 but wrote 333. It is 330:D
Lecture 53
Setting up the Unit Circle Part 1 and Reference Angle
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Setting up the Unit Circle Part 1 and Reference Angle
In a 2 part video I introduce the basic functions of Trigonometry (Sine, Cosine, and Tangent) and explain how to develop the unit circle using 45-45-90 triangle and 30-60-90 triangles you learned in Geometry. I also define reference angles.
Setting Up the Unit Circle Part 2
https://www.youtube.com/watch?v=FaZ7frx8nd8&src_vid=j5SoWzBS...
Lecture 54
Setting Up the Unit Circle Part 2
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Setting Up the Unit Circle Part 2
In a 2 part video I introduce the basic functions of Trigonometry ( Sine, Cosine, and Tangent ) and explain the design and use of the unit circle. I also review the definition of a reference angle.
Lecture 55
Linear & Angular Speed Part 1
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Linear & Angular Speed Part 1
I go over examples of evaluating Linear Speed, Angular Speed, and finding Arc Length while measuring rotation using radians. In this video I keep saying Velocity matching language of textbooks I have used in the past, but since the measure of velocity also incorporates direction I should be saying speed.
Linear & Angular Speed Part 2
https://www.youtube.com/watch?v=iMgckNT8K6s
Lecture 56
Linear & Angular Speed Part 2
Play Video
Linear & Angular Speed Part 2
I go over examples of evaluating Linear Speed, Angular Speed, and finding Arc Length while measuring rotation using radians.
In this video I keep saying Velocity matching language of textbooks I have used in the past, but since the measure of velocity also incorporates direction I should be saying speed.
Lecture 57
Evaluating Trig Functions w/ Unit Circle Degrees & Radians
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Evaluating Trig Functions w/ Unit Circle Degrees & Radians
I go over many example of evaluating trigonometry functions in exact form using the unit circle.
Lecture 58
Fundamental Trigonometric Identities Intro & Proofs
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Fundamental Trigonometric Identities Intro & Proofs
I introduce and prove the Fundamental Trigonomic Identities...the Quotient Identities, Reciprocal Identities, and the Pythagorian Identities.
Lecture 59
Trig Expressions & Finding Trig Functions Given another Trig Ratio
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Trig Expressions & Finding Trig Functions Given another Trig Ratio
I do a couple more examples of evaluating trig expressions using the unit circle. I then show you how to set up multiple Trigonometric Functions from a single angle....Say given Sine find Tangent of the same angle.
Lecture 60
Right Triangle Trigonometry Part 1: Finding Missing Sides
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Right Triangle Trigonometry Part 1: Finding Missing Sides
I introduce evaluating trigonometric functions about a right triangle. We will set up the trig ratios given a right triangle, and find missing angle measures, and sides of right triangles using the pythagorean theorm and the trig functions.
Lecture 61
Right Triangle Trigonometry Part 2: Solving for Acute Angles
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Right Triangle Trigonometry Part 2: Solving for Acute Angles
Link to Part 1 https://www.youtube.com/watch?v=pkjuVZUdcvo
I introduce evaluating trigonometric functions about a right triangle. We will set up the trig ratios given a right triangle, and find missing angle measures, and sides of right triangles using the pythagorean theorm and the trig functions.
Lecture 62
Trigonometric Cofunctions
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Trigonometric Cofunctions
I introduce the concept of Cofunctions in Trigonometry and explain why and when they are equal.
Lecture 63
Trigonometric Functions of Any Angle
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Trigonometric Functions of Any Angle
I introduce how to evaluate trig functions without knowing the actual angle measures. With just enough information to determine what quadrant an angle is in and set up the reference triangle, we can find any trigonometric ratio.
Lecture 64
Understanding Basic Sine & Cosine Graphs
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Understanding Basic Sine & Cosine Graphs
I use the unit circle to graph 2 periods the basic sine and cosine functions to show how they relate to each other. I also explain how the symmetry of these two graphs helps you to determine that the sine function is odd and the cosine function is even.
Lecture 65
Graphing Sine & Cosine w/out a Calculator Pt1
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Graphing Sine & Cosine w/out a Calculator Pt1
I introduce the transformations you can apply to the six trigonomtric functions. I then go over two basic examples of graphing sine with using t-tables. Part 2 will be two more examples of graphing sine & cosine with more transformations using a t-table. Part 2 http://www.youtube.com/watch?v=c1VD_LEs5ZY&feature=share&lis...
Lecture 66
Graphing Sine & Cosine w/out a Calculator Pt 2
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Graphing Sine & Cosine w/out a Calculator Pt 2
I introduce the transformations you can apply to the six trigonomtric functions. I then go over two basic examples of graphing sine with using t-tables. Part 2 will be two more examples of graphing sine & cosine with more transformations using a t-table.
Lecture 67
Equation of Sine and Cosine from a Graph
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Equation of Sine and Cosine from a Graph
Lecture 68
Water Depth Word Problem Modeled with Cosine Sine Function
Play Video
Water Depth Word Problem Modeled with Cosine Sine Function
Analyzing the time between high tides and the depth of the water, I make a function that models the tides and then calculate the safe hours for a ship to enter the port.

At minute 13:27 I wrote the inverse cosine of .667 or arccos(.667), that should be arccos(-.667).
Lecture 69
Intro Tangent & Cotangent Graphs
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Intro Tangent & Cotangent Graphs
Using the unit circle I explain why Tangent and Cotangent have a period of Pi instead of 2Pi like the other trig functions. I then graph the two parent functions discussing domain and range as well.
Lecture 70
Tangent & Cotangent Graphs w/ Transformations
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Tangent & Cotangent Graphs w/ Transformations
I do two examples of Tangent and Cotangent that include multiple transformations with a t-table. This video has two errors noted below which have annotation corrections in the video which you will see if you have Flash.

At minute 7:27 I say that the cot(0) is -1/0, that should be 1/0.

The very last point I mention at around 10:42 should have a y coordinate of -1 not 0. Sorry for the small errors:)
Lecture 71
Graphing Secant & Cosecant w/ t-table
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Graphing Secant & Cosecant w/ t-table
I show the reciprocal relationship between the Cosine and Secant graph and Sine Cosecant graph. This video includes two examples of graphing these inverse functions. Note the reciprocal identities CSC(theta)=1/SIN(theta) and SEC(theta)=1/COS(theta) as I work through these examples.
Lecture 72
Evaluating Inverse Trigonometric Functions
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Evaluating Inverse Trigonometric Functions
I introduce Inverse Trigonometric Functions. I explain where the restricted range values of inverse sine, inverse cosine, and inverse tangent come from...and do a number of examples.
Lecture 73
Evaluating Inverse Trigonometric Functions (Full Length PLEASE READ DESCRIPTION
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Evaluating Inverse Trigonometric Functions (Full Length PLEASE READ DESCRIPTION
I introduce Inverse Trigonometric Functions. I explain where the restricted range values of inverse sine, inverse cosine, and inverse tangent come from...and do a number of examples.

THIS VIDEO CONTAINS AN ANNOTATION CORRECTION FOR THE RANGE OF INVERSE TANGENT. Inverse Tangent has a range between -pi/2 & pi/2 but does not include these values. This should read -pi/2 is less than theta less than pi/2. I apologize for my error when copying my notes for the board.

Evaluating Inverse Trigonometric Functions Corrected
https://www.youtube.com/watch?v=7t_pZGGxMdE&list=PL085526F86...
Lecture 74
Verifying Trigonometric Identities - Part 1
Play Video
Verifying Trigonometric Identities - Part 1
Using the fundemental identities and the Pythagorean Identities, I go over multiple examples of verifying trigonometric identities.

It is very important in proofs that you do not handle it like an equation moving terms and factors from side to side.

I was corrected that what I am trying to prove should not be within the body of the proof. This implies it has already been assumed to be true. So proofs should be shown like this example:
cos^2(x)(tan^2(x)+1)=1
Proof:
cos^2(x)(tan^2(x)+1)=cos^2(x)*sec^2(x)
=cos^2(x)(1/cos^2(x))
=1

Verifying Trigonometric Identities Part 2
https://www.youtube.com/watch?v=q8k-sS7qRts
Verifying Trigonometric Identities Part 3
https://www.youtube.com/watch?v=mAnw4ImaPK0
Lecture 75
Verifying Trigonometric Identities - Part 2
Play Video
Verifying Trigonometric Identities - Part 2
Using the fundemental identities and the Pythagorean Identities, I go over multiple examples of verifying trigonometric identities.

It is very important in proofs that you do not handle it like an equation moving terms and factors from side to side.

I was corrected that what I am trying to prove should not be within the body of the proof. This implies it has already been assumed to be true. So proofs should be shown like this example:
cos^2(x)(tan^2(x)+1)=1
Proof:
cos^2(x)(tan^2(x)+1)=cos^2(x)*sec^2(x)
=cos^2(x)(1/cos^2(x))
=1
Lecture 76
Verifying Trigonometric Identities Pt3
Play Video
Verifying Trigonometric Identities Pt3
Using the fundemental identities and the Pythagorean Identities, I go over multiple examples of verifying trigonometric identities.

It is very important in proofs that you do not handle it like an equation moving terms and factors from side to side.

I was corrected that what I am trying to prove should not be within the body of the proof. This implies it has already been assumed to be true. So proofs should be shown like this example:
cos^2(x)(tan^2(x)+1)=1
Proof:
cos^2(x)(tan^2(x)+1)=cos^2(x)*sec^2(x)
=cos^2(x)(1/cos^2(x))
=1
Lecture 77
Sum and Difference Trigonometric Identities
Play Video
Sum and Difference Trigonometric Identities
Using the Sum and Difference Identities, I do examples of evaluating trigonometric expressions that require the use of the sum and difference identities for sine, cosine, and tangent.
At the end of the example of sin(195) I say the denominator of the answer is 2 but 2*2 equals 4. Sorry for the error. There is an annotation correction, but you will not see it if you do not have Flash.
Lecture 78
Verifying Trigonometric Identities Involving Sum & Difference
Play Video
Verifying Trigonometric Identities Involving Sum & Difference
I work through some examples of verifying trig identities that require the use of sum & difference identities of Sine, Cosine, and Tangent
I was corrected that what I am trying to prove should not be within the body of the proof. This implies it has already been assumed to be true. So proofs should be shown like this example:
cos^2(x)(tan^2(x)+1)=1
Proof:
cos^2(x)(tan^2(x)+1)=cos^2(x)*sec^2(x)
=cos^2(x)(1/cos^2(x))
=1
Lecture 79
Evaluating Trigonometry Expressions with Half and Double Angles Pt1
Play Video
Evaluating Trigonometry Expressions with Half and Double Angles Pt1
In a 2 part video I work many examples of evaluating trigonometric expressions involving half and double angle identities.
Lecture 80
Evaluating Trigonometry Expressions with Half and Double Angles Pt2
Play Video
Evaluating Trigonometry Expressions with Half and Double Angles Pt2
In a 2 part video I work many examples of evaluating trigonometric expressions involving half and double angle identities.
Lecture 81
Trigonometry Proofs Involving Half and Double Angles
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Trigonometry Proofs Involving Half and Double Angles
I work through 5 examples of verifying trigonometric identities that involve half angle and double angle identities.
I was corrected that what I am trying to prove should not be within the body of the proof. This implies it has already been assumed to be true. So proofs should be shown like this example:
cos^2(x)(tan^2(x)+1)=1
Proof:
cos^2(x)(tan^2(x)+1)=cos^2(x)*sec^2(x)
=cos^2(x)(1/cos^2(x))
=1
Lecture 82
Trigonometric Equations Single Angle 0 to 2pi Restriction
Play Video
Trigonometric Equations Single Angle 0 to 2pi Restriction
I do multiple examples of solving equations with trigonometric functions in them. This videos only includes trig functions with single angles such as sinx as opposed to sin(4x).
Lecture 83
Single Angle Trigonometric Equations All Solutions
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Single Angle Trigonometric Equations All Solutions
I do multiple examples of solving equations with trigonometric functions in them. This videos only includes trig functions with single angles such as sinx as opposed to sin(4x).
Lecture 84
Trigonometric Equations Multiple Angles 0 to 2pi Restriction
Play Video
Trigonometric Equations Multiple Angles 0 to 2pi Restriction
I continue my examples of solving algebraic equations that involve multiple angles.
Lecture 85
Trigonometric Equations Multiple Angles All Solutions
Play Video
Trigonometric Equations Multiple Angles All Solutions
I continue my examples of solving algebraic equations that involve multiple angles such as sin(2x).
Lecture 86
Oblique Triangles Law of Sines
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Oblique Triangles Law of Sines
I introduce the Law of Sine and go over a couple of examples where there is one unique triangle. I finish with an example of finding area of an oblique triangle.
Lecture 87
Ambiguous Case for Law of Sines
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Ambiguous Case for Law of Sines
I introduce solving oblique triangles when the information given is in the SSA form...or the ambiguous case. In this setting there may be one solution, two solutions, or none at all.

NOTE: At minute 14:37 I incorrectly stated 61*sin(39)=37.8, it is 38.4. At minute 16:50 I state that angle B is 48.4 degrees. I hit the wrong button, it is 50.2 degrees. There are annotation corrections, though they will not be visible on an iPhone or iPad. I am very sorry for the errors.
Lecture 88
Law of Cosines
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Law of Cosines
I introduce and do examples of solving oblique triangles using the law of cosine.
Lecture 89
Area of oblique triangles SAS SSS Heron's Formula
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Area of oblique triangles SAS SSS Heron's Formula
I introduce and do examples of finding the area of oblique triangles using two sides and an includeds angle, as well as using Heron's formula when you know all three sides.
Lecture 90
Applications of Law of Sines and Cosines
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Applications of Law of Sines and Cosines
I do four examples to help you understand how to solve some of your word problems that require Law of Sine and/or Cosine.
Lecture 91
Trigonometry Bearing Problems 4 Examples
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Trigonometry Bearing Problems 4 Examples
In this lesson I start out explaining how Bearing describes a direction of movement. I then work through 4 examples.
Example 1 involves Right Triangle Trigonometry SOHCATTOA at 4:24
Example 2 involves Pythagorean Theorem at 12:52
Example 3 involves Law of Cosine at 19:38
Example 4 we find a new Bearing using Law of Sine at 25:36
Right Triangle Trigonometry Part 1
https://www.youtube.com/watch?v=pkjuVZUdcvo&index=11&list=PL...
Oblique Triangles Law of Sines
https://www.youtube.com/watch?v=FtYbQ8X7U_w&list=PL085526F86...
Law of Cosines
https://www.youtube.com/watch?v=07w-wk8kRRE&index=39&list=PL...
Lecture 92
Understanding Polar Coordinates
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Understanding Polar Coordinates
I introduce Polar Coordinate System.
Lecture 93
Graphing Polar Equations, Test for Symmetry & 4 Examples Corrected
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Graphing Polar Equations, Test for Symmetry & 4 Examples Corrected
This lesson first starts with how to test for symmetry in a polar graph.
Symmetry to the Polar Axis at 1:34
Symmetry to the line Theta=pi/2 at 8:13
Symmetry to the Pole at 10:38
Special Types of Graphs
Circles at 13:13
Limacons at 16:16 I explain how to recognize the 4 subcategories of Limacons which are Inner Loop, Cartiod, Dimpled, and Convex.
EXAMPLE of graphing a Limacon with an inner loop at 20:29
Rose Curve introduced at 34:10
EXAMPLE with even number of petals at 38:01
EXAMPLE with odd number of petals at 49:36
Lemniscates introduced at 54:03
LAST EXAMPLE at 55:14

Graphing Calculator Example at 26:50

My introductory video of Understanding Polar Coordinates https://www.youtube.com/watch?v=tKi05dfUhAA
Lecture 94
Converting Coordinates between Polar and Rectangular Form
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Converting Coordinates between Polar and Rectangular Form
I do two examples converting a point from rectangular to polar...then back to rectangular form.
Lecture 95
Converting Equations Between Polar & Rectangular Form
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Converting Equations Between Polar & Rectangular Form
I work through 8 examples of converting equations between rectangular and polar form.
Graphing Ellipses and Circles
https://www.youtube.com/watch?v=Ux8gEMccP9w
Lecture 96
Complex Numbers in Polar Form
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Complex Numbers in Polar Form
I explain the relationhip between complex numbers in rectangular form and polar form. I also do an example of converting back and forth between the two forms.
Lecture 97
Product & Quotient of Polar Complex Numbers
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Product & Quotient of Polar Complex Numbers
I work through a couple of examples of multiplying and dividing complex numbers in polar form
Lecture 98
De Moivre's Theorem powers of Polar Complex Numbers
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De Moivre's Theorem powers of Polar Complex Numbers
I explain how to raise complex numbers in polar form by very high powers by using De Moivre's Theorem.

The first example starts at 6:13
Lecture 99
De Moivre's Theorem Roots of Polar Complex Numbers
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De Moivre's Theorem Roots of Polar Complex Numbers
I do three examples of finding roots of complex numbers in polar form using De Moivre's Root Theorem.
Lecture 100
Introduction to Vectors
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Introduction to Vectors
In this admittedly long winded video I introduce the basic vocabulary and concepts of vectors. Included definition/concepts are magnitude, direction, horizonal and vertical components, equal & opposite vectors, zero vector, unit vector, scaler multiple. I include examples of adding & subtracting vectors both graphically and numerically, applying a scaler multiple, and coverting vectors writing in terms of their horizontal & verical components into direction and magnitude.
Lecture 101
Writing Vector in terms of Magnitude & Direction Example
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Writing Vector in terms of Magnitude & Direction Example
I work through two examples of converting the description of a vector between Magnitude & Direction vs Horizontal & Vertical components.
Lecture 102
Vector Application Examples
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Vector Application Examples
I work through 5 examples of application of vectors. NOTE: In the last example I state the wind speed is 27 mph, but then use a wind speed of 10 mph in my problem. I added annotations over the written example but you will not see this on your iPad or iPhone.

The angle Theta in the work formula of the first example is "The difference between the angle the force is being applied and the direction of the work."

Because I am just a math teacher and not a science teacher, I learned something from a viewer/teacher I am guessing. Here is there reply...

59ejf
I liked your video.

However, a couple of comments are warranted.

First:
Force and energy are not the same thing and should not be used interchangeably. In fact, the vertical component of the force exerted on the handle of your wagon imparts no energy on the wagon. In fact, it doesn't even do any work on the wagon. Thus there is NO waste of energy.

Second:
This gets complicated. For instance, if a person is holding a couple of five pound buckets away from her body, she will soon tire and have to quit holding them up.

She is not doing any work on the buckets by holding them up, as work is force times distance. If the buckets don't move, no distance is travelled by the forces and no work is done. She is exerting a force, but the forces are static.

Energy and work have the same units. If no work is done, no energy is expended. Yet if you ask her if she did any work, she will say duh, to exhaustion.

Be careful with using humans in your physics problems.
Lecture 103
Dot Product & Angle Between Vectors
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Dot Product & Angle Between Vectors
I explain how to find the Dot Product and the properties of the dot product. I continue by explaining how to calculate the angle between to vectors, including the special cases of Parallel Vectors and Orthogonal Vectors.
Lecture 104
Projection of a Vector onto another Vector
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Projection of a Vector onto another Vector
I work through projecting a vector onto another vector in two setting: 1) When the vectors are described with magnitude and direction. 2) When the vectors are described by their horizontal and vertical components. NOTE: If you check to see if the composite vectors (at the end of this video) are perpendicular, the dot product will not equal zero. I rounded off my work too much when working through the scaler multiple portion of the projection formula.
Lecture 105
Solving Linear Systems with Substitution and Linear Combination
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Solving Linear Systems with Substitution and Linear Combination
I take a system of linear equations and show you how to solve it with substition and then with linear combination. I show how you could find one answer, no answer, or many answers.
Lecture 106
Solving 3 Variable Linear Systems Substitution / Gaussian Elimination
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Solving 3 Variable Linear Systems Substitution / Gaussian Elimination
I introduce how to solve 3 variable linear systems with the elimination method and then work through 2 examples that each have one unique solution.
Lecture 107
Applications of 3 Variable Systems
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Applications of 3 Variable Systems
I work through 2 examples of solving 3 variable linear systems using the elimination method. I show how to find the equation of a parabola when you are given 3 random points the graph goes through in my first example. In the second example I figure out how some money was split between three investments.
Lecture 108
Non-Square 3 Variable Linear Systems
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Non-Square 3 Variable Linear Systems
I work through 2 example of "solving" non square systems. When you have three unknowns and only 2 equations you will not be able to find numerical solutions, but we can learn how the three unknowns are related to each other.
Lecture 109
Partial Fraction Decomposition Part 1
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Partial Fraction Decomposition Part 1
In a 2 part video I explain Partial Fraction Decomposition. This is the process where we can undo the addition of two algebraic fractions. Part one deals with linear factors in the denominator and part two deals with quadratic factors in the denominator.
Partial Fractions Decomposition Part 2
https://www.youtube.com/watch?v=3qo527nyiaY
Lecture 110
Partial Fractions Decomposition Part 2
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Partial Fractions Decomposition Part 2
In a 2 part video I explain Partial Fraction Decomposition. This is the process where we can undo the addition of two algebraic fractions. Part one deals with linear factors in the denominator and part two deals with quadratic factors in the denominator.
Lecture 111
2 Variable Non Linear Systems Substitution Method
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2 Variable Non Linear Systems Substitution Method
I explain and work through 2 examples of solving 2 Variable Non Linear Systems with the Substitution Method.
Lecture 112
2 Variable Non Linear Systems Addition/Elimination Method
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2 Variable Non Linear Systems Addition/Elimination Method
I explain and work through 2 examples of solving 2 Variable Non Linear Systems with the Addition Method.
Lecture 113
Graphing Linear Inequalities
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Graphing Linear Inequalities
I introduce how to graph 2 variable linear inequalities and work through 3 examples.
Lecture 114
Graphing 1 Variable Inequalities
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Graphing 1 Variable Inequalities
I introduce graphing of vertical and horizontal lines and inequalities.
Lecture 115
Graphing Non-Linear Inequalities
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Graphing Non-Linear Inequalities
I introduce how to graph non-linear inequalities and then finish by working through three examples.
Lecture 116
Graphing System of Linear Inequalities
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Graphing System of Linear Inequalities
I work through two examples of graphing systems of inequalities. The first example involve linear inequalities and the second example has non-linear inequalities.
Lecture 117
Graphing Ellipses & Circles
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Graphing Ellipses & Circles
I introduce the 4 conic sections, explain the basic structure of Ellipses & Circles, and finish by working through three graphing examples after putting these equations in standard form.
Lecture 118
Graphing Hyperbolas in Standard Form
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Graphing Hyperbolas in Standard Form
I introduce the basic structure of hyperbolas discussing how to locate the vertices, foci, transverse axis, conjugate axis, asymptotes, etc. I finish by working through multiple examples.
Lecture 119
Graphing Parabolas in Standard Form
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Graphing Parabolas in Standard Form
I introduce the basic structure of Parabolas. After the introductory notes, I work through three examples from easy to the hardest example last. These examples are put into Standard Form before graphing by completing the square when needed.
Lecture 120
Application of Ellipses
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Application of Ellipses
I work through three example of Applications of Ellipses.
Lecture 121
Application of Hyperbolas
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Application of Hyperbolas
I work through 2 examples of Application of Hyperbolas
Lecture 122
Applications of Parabolas in Standard Form
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Applications of Parabolas in Standard Form
I work through 2 examples of word problems involving parabolas.
Lecture 123
Rotated Conic Section Identifying & Graphing 4 Examples
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Rotated Conic Section Identifying & Graphing 4 Examples
Two examples of Identifying Rotated Conics at 5:53
Graphing a Conic Rotated 30 degrees at 15:19
Graphing a Rotated Conic with your Graphing calculator at 28:55
Graphing a Conic with a non unit circle angle of rotation and center not on the origin at 36:27
Lecture 124
Introduction to Parametric Equations
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Introduction to Parametric Equations
I introduce the basic concepts of Parametric Equations. I then work through many examples of graphing with t-tables.
Lecture 125
Parametric Equations Eliminating Parameter T
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Parametric Equations Eliminating Parameter T
I do multiple examples of converting parametric equations into rectangular form by eliminating the parameter T.
Lecture 126
Intro to Sequences
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Intro to Sequences
I introduce the concept and notation for sequences and finish with a couple of graphing examples. I then explain recursive formulas and do another 2 examples showing you how they work.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 127
Summation Notation (Sigma)
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Summation Notation (Sigma)
I introduce the Summation Notation...Sigma...and work through five examples related to the topic of sequences.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 128
Factorial Notation
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Factorial Notation
I introduce the Factorial Notation or symbol. I then work through a number of examples of how to evaluate factorials, and examples of factorials in sequence notation and summation notation.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 129
Arithmetic Sequence Introduction
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Arithmetic Sequence Introduction
I introduce the concept of Arithmetic Sequences and work through some examples.

For all my lessons so far on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 130
Partial Sum Arithmetic Sequence
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Partial Sum Arithmetic Sequence
I introduce the formula for finding the sum of 'n' terms of an arithmetic sequence. I then work through several examples.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 131
Intro to Geometric Sequences
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Intro to Geometric Sequences
I introduce the concept of Geometric Sequences and writing the general formula for this type of sequence. I finish by working through several examples.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 132
Geometric Means in a Geometric Sequence
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Geometric Means in a Geometric Sequence
I explain how to find missing Geometric Means within a Geometric Sequence. I finish with two examples...first one relatively easy...the second is not so much:)
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 133
Partial Sum Geometric Sequences
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Partial Sum Geometric Sequences
I introduce the formula for finding the sum of 'n' terms of an geometric sequence. I then work through several examples.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 134
Infinite Sum Geometric Series
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Infinite Sum Geometric Series
I introduce the formula for finding the Infinite Sum of a Geometric Series, explain briefly how it works, and I then work through multiple examples.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 135
Converting Repeating Decimal to a Fraction
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Converting Repeating Decimal to a Fraction
I explain how finding the sum of an infinite geometric series allows you to convert a repeating decimal into a fraction.
For all my lessons on Sequences and Series go here:)
http://www.profrobbob.com/pre-calculus/intro-to-sequences-ar...
Lecture 136
Induction Proofs
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Induction Proofs
I introduce the steps to doing an Induction Proof. I then work through three examples that I hope will help you understand the concept and complete your homework:)
Lecture 137
Binomial Theorem Introduction to Raise Binomials to High Powers
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Binomial Theorem Introduction to Raise Binomials to High Powers
I explain how Pascal's Triangle and the Binomial Theorem help you to quickly expand binomials raised to relatively high powers. My last example is using the Binomial Theorem to find a specific term within an expansion.
Lecture 138
Why Limits are Important in Calculus
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Why Limits are Important in Calculus
I give an overview of how limits are used in two main concepts of Calculus. Finding slope of a curve at a given point and finding area under a curve.
Lecture 139
Finding Real limits Graphical & Numerical Approach
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Finding Real limits Graphical & Numerical Approach
I work through five examples of finding REAL limits by analyzing graphs and numerical tables. Infinite limits are not discussed in this video. EXAMPLES AT 2:51 5:17 8:45 12:00 14:03

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Lecture 140
Properties of Limits
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Properties of Limits
I introduce the properties of limits that allow you to find the real limit of a function as x approaches "a" without needing to make a t-table or graph the function.

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Lecture 141
Finding Limits with Properties Examples inc Quotients
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Finding Limits with Properties Examples inc Quotients
I work through 6 examples of finding the limit of a function as "x" approaches "a"...though at the start I state that I am doing 4:\ Four of these examples involve quotients where the function may be undefined at "a". EXAMPLES AT 2:23 3:20 6:05 8:23 13:40 18:59

Please Like, and Follow as well at:
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Lecture 142
Limits & Continuity
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Limits & Continuity
I explain how to test for Continuity at a given point and find Discontinuities in functions. I finish by working through a couple of examples. EXAMPLES 4:57 10:57. Though I say smooth and continuous a few times in this video, a function does not have to be smooth to be continuous.
Lecture 143
Limits of Piecewise Function
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Limits of Piecewise Function
I work through four examples of finding limits of piecewise functions using the properties of limits. EXAMPLES 0:20 5:57 10:50 15:10

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Lecture 144
Slope of Tangent Line Derivative at a Point
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Slope of Tangent Line Derivative at a Point
I explain how to find the instantaneous slope at a given point along a curve. I finish by finding the tangent line to that curve at that point. All three of the examples I work in this video have real values of slope and thus have a real limit as h approaches 0. I do examples that have vertical tangent lines at the point of tangency and thus an undefined slope in a later video. EXAMPLES at 8:39 13:11 20:21

Link to Full Length Introduction to Definition of Derivative and Tangent Line Problems.
http://www.youtube.com/watch?v=eWxo7O506G8&feature=share&lis...
Lecture 145
Finding Derivative with Definition of Derivative
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Finding Derivative with Definition of Derivative
I work through two examples of finding the derivative of a function using the Definition of Derivatives.