  # Trigonometry, Polar Coordinates, & Complex Numbers

## Video Lectures

Displaying all 55 video lectures.
 Lecture 1 Play Video Standard Position Angles & Radians Part 1Part 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 2 Play Video Standard Position Angles & Radians Part 2In 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 3 Play Video Angle Measures in Degrees Minutes & Seconds DMS Lecture 4 Play Video Setting up the Unit Circle Part 1 and Reference AngleIn 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 5 Play Video Setting Up the Unit Circle Part 2In 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 6 Play Video Linear & Angular Speed Part 1I 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 7 Play Video Linear & Angular Speed Part 2I 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 8 Play Video Evaluating Trig Functions w/ Unit Circle Degrees & RadiansI go over many example of evaluating trigonometry functions in exact form using the unit circle. Lecture 9 Play Video Fundamental Trigonometric Identities Intro & ProofsI introduce and prove the Fundamental Trigonomic Identities...the Quotient Identities, Reciprocal Identities, and the Pythagorian Identities. Lecture 10 Play Video Trig Expressions & Finding Trig Functions Given another Trig RatioI 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 11 Play Video Right Triangle Trigonometry Part 1I 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 12 Play Video Right Triangle Trigonometry Part 2Link 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 13 Play Video Trigonometric CofunctionsI introduce the concept of Cofunctions in Trigonometry and explain why and when they are equal. Lecture 14 Play Video Trigonometric Functions of Any AngleI 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 15 Play Video Understanding Basic Sine & Cosine GraphsI 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 16 Play Video Graphing Sine & Cosine w/out a Calculator Pt1I 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 17 Play Video Graphing Sine & Cosine w/out a Calculator Pt 2I 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 18 Play Video Equation of Sine and Cosine from a Graph Lecture 19 Play Video Water Depth Word Problem Modeled with Cosine Sine FunctionAnalyzing 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 20 Play Video Intro Tangent & Cotangent GraphsUsing the unit circle I explain why Tangent and Cotangent have a period of π instead of 2π like the other trig functions. I then graph the two parent functions discussing domain and range as well. Lecture 21 Play Video Tangent & Cotangent Graphs w/ TransformationsI 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 22 Play Video Graphing Secant & Cosecant w/ t-tableI 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 23 Play Video Evaluating Inverse Trigonometric FunctionsI 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 24 Play Video Verifying Trigonometric Identities Pt 1Using 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)) =1Verifying 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 25 Play Video Verifying Trigonometric Identities - Part IUsing 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 I reshot the intro with permanent corrections instead of just the annotations that are at the beginning of this video. http://youtu.be/TCdhf9iVkYcVerifying 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 26 Play Video Verifying Trigonometric Identities - Part IIUsing 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 27 Play Video Verifying Trigonometric Identities - Part IIIUsing 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 28 Play Video Sum and Difference Trigonometric IdentitiesUsing 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 29 Play Video Verifying Trigonometric Identities Involving Sum & DifferenceI 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 30 Play Video Evaluating Trigonometry Expressions with Half and Double Angles Pt1In a 2 part video I work many examples of evaluating trigonometric expressions involving half and double angle identities. Lecture 31 Play Video Evaluating Trigonometry Expressions with Half and Double Angles Pt2In a 2 part video I work many examples of evaluating trigonometric expressions involving half and double angle identities. Lecture 32 Play Video Trigonometry Proofs Involving Half and Double AnglesI 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 33 Play Video Trigonometric Equations Single Angle 0 to 2π RestrictionI 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 34 Play Video Single Angle Trigonometric Equations All SolutionsI 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 35 Play Video Trigonometric Equations Multiple Angles 0 to 2π RestrictionI continue my examples of solving algebraic equations that involve multiple angles. Lecture 36 Play Video Trigonometric Equations Multiple Angles All SolutionsI continue my examples of solving algebraic equations that involve multiple angles such as sin(2x). Lecture 37 Play Video Oblique Triangles Law of SinesI 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 38 Play Video Ambiguous Case for Law of SinesI 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 39 Play Video Law of CosinesI introduce and do examples of solving oblique triangles using the law of cosine. Lecture 40 Play Video Area of oblique triangles SAS SSS Heron's FormulaI 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 41 Play Video Applications of Law of Sines and CosinesI do four examples to help you understand how to solve some of your word problems that require Law of Sine and/or Cosine. Lecture 42 Play Video Understanding Polar CoordinatesI introduce Polar Coordinate System. Lecture 43 Play Video Converting Coordinates between Polar and Rectangular FormI do two examples converting a point from rectangular to polar...then back to rectangular form. Lecture 44 Play Video Converting Equations Between Polar & Rectangular FormI work through 8 examples of converting equations between rectangular and polar form. Lecture 45 Play Video Graphing Polar Equations, Test for Symmetry & 4 Examples CorrectedThis 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 55:14Graphing Calculator Example at 26:50My introductory video of Understanding Polar Coordinates https://www.youtube.com/watch?v=tKi05dfUhAA Lecture 46 Play Video Complex Numbers in Polar FormI 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 47 Play Video Product & Quotient of Polar Complex NumbersI work through a couple of examples of multiplying and dividing complex numbers in polar form Lecture 48 Play Video De Moivre's Theorem powers of Polar Complex NumbersI 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 49 Play Video De Moivre's Theorem Roots of Polar Complex NumbersI do three examples of finding roots of complex numbers in polar form using De Moivre's Root Theorem. Lecture 50 Play Video Introduction to VectorsIn 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 51 Play Video Writing Vector in terms of Magnitude & Direction ExampleI work through two examples of converting the description of a vector between Magnitude & Direction vs Horizontal & Vertical components. Lecture 52 Play Video Vector Application ExamplesI 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 53 Play Video Dot Product & Angle Between VectorsI 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 54 Play Video Projection of a Vector onto another VectorI 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 55 Play Video Trigonometry Bearing Problems - 4 ExamplesIn 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...