Unit 1 Lesson 1 - In this video I review the three primary trig functions, sine, cosine, and tangent, and use them solve right triangles. I then introduce the cofunction identity sin x = cos(90 - x) and cos x = sin(90 - x). I failed to mention that this works for all the other cofunction pairs, tan and cot, and sec and csc.
Unit 1 Lesson 3 - In this video I define a radian. I then explain how to convert between radians and degrees and degrees and radians. Finally, I demonstrate how to find arc length using S = r(theta).
Unit 1 Lesson 4 - In this video I set up the Unit circle for both degree measure and radian. I then review the two common reference triangles and use both to evaluate the 6 trig functions for that angle. I discuss the concept of coterminal angles and the effect of sign on the trig functions.
Unit 1 Lesson 5 - In this video I review the two special right triangles, the 45 45 90 and the 30 60 90 triangles. I then use them to evaluate the six trig functions on the angle 5pi/6.
Unit 1 Lesson 6 - In this video I demonstrate how to use the unit circle to find the trig functions for any reference angle in any quadrant including the boundaries between the quadrants.
Unit 2 Lesson 1 - In this video I use the unit circle to create a rectangular graph of the sine and tangent functions.
Unit 2 Lesson 2 - In this video I use Desmos to explore the domain and range of the six trigonometric functions. Of particular importance is recognizing where division by zero produces asymptotes in tan x, cot x, csc x and sec x.
Unit 2 Lesson 3 - In this video I use Desmos to explore and determine period, amplitude, even/odd, asymptotes, and zeros of sine, tangent, and cosecant.
Unit 2 Lesson 4 - In this video I explore ways to change the period, amplitude, phase, and vertical shift of the trig functions.
Unit 2 Lesson 5 - In this video I take a given sine function with modifiers and determine the function's amplitude, period, phase shift and zeros. I then take a given tangent function with modifiers and determine the function's period, phase shift, zeros and asymptotes.
Unit 2 Lesson 6 - In this video I start with a graph of a trigonometric function. I then determine the specific function that matches the graph by determining the graph's midline, amplitude, period and phase shift.
Unit 2 Lesson 7 - In this video I take a table of values, graph them, and then fit a sinusoidal function to the data. I then demonstrate some of the curve fitting features of Desmos and demonstrate that a sinusoidal function does appear to fit the given data best.
Unit 2 Lesson 8 - In this video I review what an inverse function is, the definition and techniques for finding the inverse function given a formula or a table. I then apply these concepts to inverse sine, cosine and tangent, noting the necessity of domain restrictions so that the inverse function is truly a function, or restricting the domain on a trig function so that it passes the horizontal line test. Finally, I demonstrate how to find the specific value of the inverse function with given input.
Unit 2 Lesson 9 - In this video we review the concept of composition of functions. We then discuss the implications regarding domain and range of composed functions. Then we verify inverse trig functions using right triangles in the first quadrant. Finally, we use these triangles to rewrite composition of trig functions with non-matching inverse trig functions.
Unit 2 Lesson 10 - In this video I demonstrate mathematical modelling of the musical note, Concert A4 = 440 hz. I then demonstrate the basic idea behind signal processing, called Fourier transformation, as addition of harmonic frequencies with variable coefficients. This technique enables all kinds of signal processing, including creating very realistic synthesized sounds.
Unit 3 Lesson 1 - In this video I present the reciprocal, rational and Pythagorean trigonometric identities. I then use them to prove csc x sec x = tan x + cot x.
Unit 3 Lesson 2 - In this video I derive the angle addition formulas for sine and cosine. Though you are not expected to derive them, it is useful to see where they come from, and being able to derive them is an excellent mathematical exercise.
Unit 3 Lesson 2 Part 2 - In this video I demonstrate how to use the addition formulas to find sin 75 and cos pi/12.
Unit 3 Lesson 2 Part 3 - In this video I prove the tangent angle addition identity. I then use it to find tan 7pi/12.
Unit 3 Lesson 3 - In this video I derive the double angle and half angle identities for sine, cosine and tangent.
Unit 3 Lesson 3 Part 2 - In this video I demonstrate how to use the half angle formulas to find sin 15 and tan 15.
Unit 3 Lesson 4 - In this video I derive the Law of Sines.
Unit 3 Lesson 4 Part 2 - In this video I demonstrate how to solve a non-right triangle of the type AAS or ASA using the Law of Sines.
Unit 3 Lesson 5 - In this video I derive the Law of Cosines.
Unit 3 Lesson 5 Part 2 - In this video I solve a non-right triangle of the type SAS using the Law of Cosines and the Law of Sines.
Unit 3 Lesson 5 Part 3 - In this video I solve a non-right triangle of the type SSS using the Law of Cosines rearranged for the missing angle.
Unit 3 Lesson 6 - In this video I solve a trigonometric equation by squaring both sides, using a Pythagorean identity to eliminate a trig function, solve a quadratic equation by y-substitution and eliminate an extraneous solution.
Unit 3 Lesson 6 Part 2 - In this video I solve a trigonometric equation by substitution from the Pythagorean identity for tangent and solving a quadratic equation by y-substitution.
Unit 3 Lesson 6 Part 3 - In this video I find the solution to a problem of rectilinear motion by solving a trigonometric equation by inverse operations.
Unit 4 Lesson 1 - In this video I introduce the concept of polar coordinates. I then explain how to convert polar to rectangular and rectangular to polar coordinates. Finally, I demonstrate how to plot polar coordinates in Desmos.
Unit 4 Lesson 2 - In this video I graph r = sin theta by making a t-table and calculating r values from given theta values from 0 to 2pi.
Unit 4 Lesson 4 - In this video I demonstrate the formula for graphing conic sections in polar coordinates. I convert on example to rectangular coordinates to confirm that it works. I then use Desmos to demonstrate the various forms of the formula, noting what happens to the focus, vertex, and directrix in various cases.
Unit 4 Lesson 5 - In this video I demonstrate the class of functions known as limacons, of the type r = a + b sin theta. I explore what happens when sine is replaced with cosine, the + by a -, and a and b are allowed to vary.
Unit 4 Lesson 6 - In this video I use Desmos to explore the graph of rose curves (r = a sin(b theta)) and lemniscates (r^2 = a^2 sin(2 theta)). I examine the effect of changing a and b and switching between sine and cosine.
Unit 4 Lesson 7 - In this video I demonstrate graphically, polar coordinates in rectangular and polar form. I use trigonometry to outline the important formulas and then convert from rectangular to polar and from polar to rectangular form.
Unit 4 Lesson 8 - In this video I derive the rule for multiplying complex numbers in polar form. I then present the corollaries, dividing and raising to a power. Finally, I demonstrate how to find z^10 using the rule for powers.
Unit 4 Lesson 9 - In this video I demonstrate DeMoivre's theorem for raising complex numbers to powers, but particularly focus on roots. I demonstrate how to find the cube roots of z = 3 + 4i. I then demonstrate how to find the three cube roots of z = 8, explaining how there are always n nth roots of any number, real or complex. Finally, I demonstrate how cubing one of the complex cube roots of 8 results in 8.
Unit 5 Lesson 1 - In this video I explain what a vector is, and how to represent it on a rectangular graph. I demonstrate how to find the component form, its magnitude, its direction, and how to tell if two vectors are equal.
Unit 5 Lesson 2 - In this video I demonstrate how to use the "tail-to-head" or "parallelogram" method for adding vectors in rectangular form. I then demonstrate scalar multiplication and subtraction in component form by finding 5u-2v.
Unit 5 Lesson 3 - In this video I demonstrate how to write vectors in the i, j form, where i = <1,0>, and j = <0,1>. I then demonstrate how to do vector addition, subtraction and scalar multiplication in this form. Finally, I demonstrate how to find the unit vector, that is a vector that moves in the same direction but having a magnitude of 1.
Unit 5 Lesson 4 - In this video I demonstrate writing vectors in rectangular form given the magnitude and direction. I then find the sum of two vectors by first converting them to rectangular form. I then demonstrate how to find the direction of a given vector in rectangular form.
Unit 5 Lesson 4 Part 2 - In this video I finish the example started in the previous video by finding the angle angle magnitude of the resultant vector. I then demonstrate the validity of the method of converting to rectangular form, operating, and converting to polar form by solving the problem using the law of cosines and the law of sines.
Unit 5 Lesson 5 - In this video I demonstrate how to find a dot product. I then demonstrate the relation v*v = |v|^2. Finally, I demonstrate how u*v = 0 if and only if u is perpendicular to v.
Unit 5 Lesson 6 - In this video I present the formula for the angles between two vectors as cos theta = u*v / (|u||v|). I then use it to find the angle between two vectors and then explain how to tell if two vectors are parallel, orthogonal or neither.
Unit 5 Lesson 6 Part 2 - In this video I explain the Physics definition of work as force in the direction of motion times the displacement in that direction. Combining that with the formula for this unit, we end up with W = |F||AB|cos theta = |F|*|AB|. I then use the formula to find work given angular vectors and then given rectangular vectors.
Unit 6 Lesson 1 - In this video I model the flight of a golf ball, struck at an angle of 30 degrees to the ground, with an initial velocity of 150 ft/s by writing a parametric equation in the x-y plane. I demonstrate the flight of the ball by animation in Desmos.
Unit 6 Lesson 2 - In this video I continue the previous problem by eliminating the t parameter by substitution to arrive at an equation for x and y alone. The result is a quadratic equation that matches the flight path of the ball. Remember that time information is lost when you do this.
Unit 6 Lesson 3 - In this video I demonstrate how to use the Pythagorean identities, sin^2 + cos^2 = 1, tan^2 + 1 = sec^2, and cot^2 + 1 = csc^2, to eliminate the t variable in some cases. I use Desmos to verify the results.
Unit 6 Lesson 4 - In this video I demonstrate how replacing t with a function of t does not change the path of a parametric equation, but, rather, changes the time information, starting point and rate of traversing the path. I then demonstrate how substituting a function, x(t) for x in a rectangular equation creates a parametric equation that traces the same path as the rectangular equation.
Unit 6 Lesson 5 - In this video I convert a polar equation into a parametric equation on theta. I then convert a parametric equation on theta into a polar equation. Finally, I convert a parametric function on t into a polar equation on theta. The second one involves a rather lengthy trigonometric simplification.
Unit 6 Lesson 6 - In this video I represent the motion of an object with a known starting rectangular point under constant velocity in a given direction with a parametric formula. I then use this to determine its location after 5 seconds.
Unit 6 Lesson 6 Part 2 - In this video I model the flight of a golf ball struck off the deck of a cruise ship with a parametric equation.
Unit 7 Lesson 1 - In this video I demonstrate how to find a limit using a table in Desmos, and by algebraic manipulation to remove a removable discontinuity. I explain what a left limit and a right limit are, and what a two-sided limit is.
Unit 7 Lesson 2 - In this video I demonstrate the properties of limits, particularly that they break up over addition, subtraction, multiplication, and division.
Unit 7 Lesson 3 - In this video I use the piecewise function f(x) = 1/2 x - 2 for x<= -2 and f(x) = (-x^2 + 2)(x - 2) / (x - 2) for x > 2 to demonstrate the limit definition of a continuous function at a point, name that the left limit, right limit, and function value must all be in agreement. I briefly address the case of a bounded domain [c,d].
Unit 7 Lesson 4 - In this video I begin to discuss the concept of a derivative by examining the function f(x) = x^3 - 2x^2 and looking at what it takes to find the instantaneous slope, or the slope of the tangent line at some point (a,f(a)) on f(x). I define a second point and calculate the slope of the secant line between the two points and illustrate what happens as the points move close together.
Unit 7 Lesson 4 Part 2 - In this video I find the derivative of f(x) at x = a and demonstrate how it allows me to find the equation of the tangent line at that point.
Unit 1 Lesson 3 - In this video I define a radian. I then explain how to convert between radians and degrees and degrees and radians. Finally, I demonstrate how to find arc length using S = r(theta).
Unit 1 Lesson 4 - In this video I set up the Unit circle for both degree measure and radian. I then review the two common reference triangles and use both to evaluate the 6 trig functions for that angle. I discuss the concept of coterminal angles and the effect of sign on the trig functions.
Unit 1 Lesson 5 - In this video I review the two special right triangles, the 45 45 90 and the 30 60 90 triangles. I then use them to evaluate the six trig functions on the angle 5pi/6.
Unit 1 Lesson 6 - In this video I demonstrate how to use the unit circle to find the trig functions for any reference angle in any quadrant including the boundaries between the quadrants.
Unit 2 Lesson 1 - In this video I use the unit circle to create a rectangular graph of the sine and tangent functions.
Unit 2 Lesson 2 - In this video I use Desmos to explore the domain and range of the six trigonometric functions. Of particular importance is recognizing where division by zero produces asymptotes in tan x, cot x, csc x and sec x.
Unit 2 Lesson 3 - In this video I use Desmos to explore and determine period, amplitude, even/odd, asymptotes, and zeros of sine, tangent, and cosecant.
Unit 2 Lesson 4 - In this video I explore ways to change the period, amplitude, phase, and vertical shift of the trig functions.
Unit 2 Lesson 5 - In this video I take a given sine function with modifiers and determine the function's amplitude, period, phase shift and zeros. I then take a given tangent function with modifiers and determine the function's period, phase shift, zeros and asymptotes.
Unit 2 Lesson 6 - In this video I start with a graph of a trigonometric function. I then determine the specific function that matches the graph by determining the graph's midline, amplitude, period and phase shift.
Unit 2 Lesson 7 - In this video I take a table of values, graph them, and then fit a sinusoidal function to the data. I then demonstrate some of the curve fitting features of Desmos and demonstrate that a sinusoidal function does appear to fit the given data best.
Unit 2 Lesson 8 - In this video I review what an inverse function is, the definition and techniques for finding the inverse function given a formula or a table. I then apply these concepts to inverse sine, cosine and tangent, noting the necessity of domain restrictions so that the inverse function is truly a function, or restricting the domain on a trig function so that it passes the horizontal line test. Finally, I demonstrate how to find the specific value of the inverse function with given input.
Unit 2 Lesson 9 - In this video we review the concept of composition of functions. We then discuss the implications regarding domain and range of composed functions. Then we verify inverse trig functions using right triangles in the first quadrant. Finally, we use these triangles to rewrite composition of trig functions with non-matching inverse trig functions.
Unit 2 Lesson 10 - In this video I demonstrate mathematical modelling of the musical note, Concert A4 = 440 hz. I then demonstrate the basic idea behind signal processing, called Fourier transformation, as addition of harmonic frequencies with variable coefficients. This technique enables all kinds of signal processing, including creating very realistic synthesized sounds.
Unit 3 Lesson 1 - In this video I present the reciprocal, rational and Pythagorean trigonometric identities. I then use them to prove csc x sec x = tan x + cot x.
Unit 3 Lesson 2 - In this video I derive the angle addition formulas for sine and cosine. Though you are not expected to derive them, it is useful to see where they come from, and being able to derive them is an excellent mathematical exercise.
Unit 3 Lesson 2 Part 2 - In this video I demonstrate how to use the addition formulas to find sin 75 and cos pi/12.
Unit 3 Lesson 2 Part 3 - In this video I prove the tangent angle addition identity. I then use it to find tan 7pi/12.
Unit 3 Lesson 3 - In this video I derive the double angle and half angle identities for sine, cosine and tangent.
Unit 3 Lesson 3 Part 2 - In this video I demonstrate how to use the half angle formulas to find sin 15 and tan 15.
Unit 3 Lesson 4 - In this video I derive the Law of Sines.
Unit 3 Lesson 4 Part 2 - In this video I demonstrate how to solve a non-right triangle of the type AAS or ASA using the Law of Sines.
Unit 3 Lesson 5 - In this video I derive the Law of Cosines.
Unit 3 Lesson 5 Part 2 - In this video I solve a non-right triangle of the type SAS using the Law of Cosines and the Law of Sines.
Unit 3 Lesson 5 Part 3 - In this video I solve a non-right triangle of the type SSS using the Law of Cosines rearranged for the missing angle.
Unit 3 Lesson 6 - In this video I solve a trigonometric equation by squaring both sides, using a Pythagorean identity to eliminate a trig function, solve a quadratic equation by y-substitution and eliminate an extraneous solution.
Unit 3 Lesson 6 Part 2 - In this video I solve a trigonometric equation by substitution from the Pythagorean identity for tangent and solving a quadratic equation by y-substitution.
Unit 3 Lesson 6 Part 3 - In this video I find the solution to a problem of rectilinear motion by solving a trigonometric equation by inverse operations.
Unit 4 Lesson 1 - In this video I introduce the concept of polar coordinates. I then explain how to convert polar to rectangular and rectangular to polar coordinates. Finally, I demonstrate how to plot polar coordinates in Desmos.
Unit 4 Lesson 2 - In this video I graph r = sin theta by making a t-table and calculating r values from given theta values from 0 to 2pi.
Unit 4 Lesson 4 - In this video I demonstrate the formula for graphing conic sections in polar coordinates. I convert on example to rectangular coordinates to confirm that it works. I then use Desmos to demonstrate the various forms of the formula, noting what happens to the focus, vertex, and directrix in various cases.
Unit 4 Lesson 5 - In this video I demonstrate the class of functions known as limacons, of the type r = a + b sin theta. I explore what happens when sine is replaced with cosine, the + by a -, and a and b are allowed to vary.
Unit 4 Lesson 6 - In this video I use Desmos to explore the graph of rose curves (r = a sin(b theta)) and lemniscates (r^2 = a^2 sin(2 theta)). I examine the effect of changing a and b and switching between sine and cosine.
Unit 4 Lesson 7 - In this video I demonstrate graphically, polar coordinates in rectangular and polar form. I use trigonometry to outline the important formulas and then convert from rectangular to polar and from polar to rectangular form.
Unit 4 Lesson 8 - In this video I derive the rule for multiplying complex numbers in polar form. I then present the corollaries, dividing and raising to a power. Finally, I demonstrate how to find z^10 using the rule for powers.
Unit 4 Lesson 9 - In this video I demonstrate DeMoivre's theorem for raising complex numbers to powers, but particularly focus on roots. I demonstrate how to find the cube roots of z = 3 + 4i. I then demonstrate how to find the three cube roots of z = 8, explaining how there are always n nth roots of any number, real or complex. Finally, I demonstrate how cubing one of the complex cube roots of 8 results in 8.
Unit 5 Lesson 1 - In this video I explain what a vector is, and how to represent it on a rectangular graph. I demonstrate how to find the component form, its magnitude, its direction, and how to tell if two vectors are equal.
Unit 5 Lesson 2 - In this video I demonstrate how to use the "tail-to-head" or "parallelogram" method for adding vectors in rectangular form. I then demonstrate scalar multiplication and subtraction in component form by finding 5u-2v.
Unit 5 Lesson 3 - In this video I demonstrate how to write vectors in the i, j form, where i = <1,0>, and j = <0,1>. I then demonstrate how to do vector addition, subtraction and scalar multiplication in this form. Finally, I demonstrate how to find the unit vector, that is a vector that moves in the same direction but having a magnitude of 1.
Unit 5 Lesson 4 - In this video I demonstrate writing vectors in rectangular form given the magnitude and direction. I then find the sum of two vectors by first converting them to rectangular form. I then demonstrate how to find the direction of a given vector in rectangular form.
Unit 5 Lesson 4 Part 2 - In this video I finish the example started in the previous video by finding the angle angle magnitude of the resultant vector. I then demonstrate the validity of the method of converting to rectangular form, operating, and converting to polar form by solving the problem using the law of cosines and the law of sines.
Unit 5 Lesson 5 - In this video I demonstrate how to find a dot product. I then demonstrate the relation v*v = |v|^2. Finally, I demonstrate how u*v = 0 if and only if u is perpendicular to v.
Unit 5 Lesson 6 - In this video I present the formula for the angles between two vectors as cos theta = u*v / (|u||v|). I then use it to find the angle between two vectors and then explain how to tell if two vectors are parallel, orthogonal or neither.
Unit 5 Lesson 6 Part 2 - In this video I explain the Physics definition of work as force in the direction of motion times the displacement in that direction. Combining that with the formula for this unit, we end up with W = |F||AB|cos theta = |F|*|AB|. I then use the formula to find work given angular vectors and then given rectangular vectors.
Unit 6 Lesson 1 - In this video I model the flight of a golf ball, struck at an angle of 30 degrees to the ground, with an initial velocity of 150 ft/s by writing a parametric equation in the x-y plane. I demonstrate the flight of the ball by animation in Desmos.
Unit 6 Lesson 2 - In this video I continue the previous problem by eliminating the t parameter by substitution to arrive at an equation for x and y alone. The result is a quadratic equation that matches the flight path of the ball. Remember that time information is lost when you do this.
Unit 6 Lesson 3 - In this video I demonstrate how to use the Pythagorean identities, sin^2 + cos^2 = 1, tan^2 + 1 = sec^2, and cot^2 + 1 = csc^2, to eliminate the t variable in some cases. I use Desmos to verify the results.
Unit 6 Lesson 4 - In this video I demonstrate how replacing t with a function of t does not change the path of a parametric equation, but, rather, changes the time information, starting point and rate of traversing the path. I then demonstrate how substituting a function, x(t) for x in a rectangular equation creates a parametric equation that traces the same path as the rectangular equation.
Unit 6 Lesson 5 - In this video I convert a polar equation into a parametric equation on theta. I then convert a parametric equation on theta into a polar equation. Finally, I convert a parametric function on t into a polar equation on theta. The second one involves a rather lengthy trigonometric simplification.
Unit 6 Lesson 6 - In this video I represent the motion of an object with a known starting rectangular point under constant velocity in a given direction with a parametric formula. I then use this to determine its location after 5 seconds.
Unit 6 Lesson 6 Part 2 - In this video I model the flight of a golf ball struck off the deck of a cruise ship with a parametric equation.
Unit 7 Lesson 1 - In this video I demonstrate how to find a limit using a table in Desmos, and by algebraic manipulation to remove a removable discontinuity. I explain what a left limit and a right limit are, and what a two-sided limit is.
Unit 7 Lesson 2 - In this video I demonstrate the properties of limits, particularly that they break up over addition, subtraction, multiplication, and division.
Unit 7 Lesson 3 - In this video I use the piecewise function f(x) = 1/2 x - 2 for x<= -2 and f(x) = (-x^2 + 2)(x - 2) / (x - 2) for x > 2 to demonstrate the limit definition of a continuous function at a point, name that the left limit, right limit, and function value must all be in agreement. I briefly address the case of a bounded domain [c,d].
Unit 7 Lesson 4 - In this video I begin to discuss the concept of a derivative by examining the function f(x) = x^3 - 2x^2 and looking at what it takes to find the instantaneous slope, or the slope of the tangent line at some point (a,f(a)) on f(x). I define a second point and calculate the slope of the secant line between the two points and illustrate what happens as the points move close together.
Unit 7 Lesson 4 Part 2 - In this video I find the derivative of f(x) at x = a and demonstrate how it allows me to find the equation of the tangent line at that point.