The Trigonometry and Pre-Calculus Tutor - 2 DVD Set! - 5 Hour Course!
by Jason Gibson
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DVD details
Actor: Jason GibsonDirector: Jason Gibson
DVD: Region Code 1
Audio: English (Original Language)
Format: Closed-captioned, Color, DVD, Full Screen, NTSC
Picture Format: 1.33:1
Running Time: 300 minutes
DVD Release Date: 2005-08-17
Audience Rating: NR (Not Rated)
Studio: Tapeworm Video
DVD Reviews of The Trigonometry and Pre-Calculus Tutor - 2 DVD Set! - 5 Hour Course!
DVD Review: Competent Marker Board Lectures On Some Rote Problems
Summary: 4 Stars
The DVDs in "The Trigonometry And Pre-Calculus Tutor" show Jason Gibson solving mathematical problems from those courses. Jason works the "crank-turning" type of problem. He doesn't explain the conceptual basis for the mathematics. (This isn't like a high school math class taught by Dr. Millbee; it's like a class taught by Coach Glonther.) Jason's terminology may originate with secondary school teachers who are desperate that their students acquire some capability with the material, but some of it is not correct mathematical terminology. For example, he promotes a concept of "opposites" instead of speaking precisely about "inverse functions".
The entire video uses one camera angle, which shows Jason working at the marker board. There are no fancy computer graphics to illustrate mathematical ideas. Jason is well prepared. He speaks clearly. His presentation is spontaneous. It is also very organized and smooth. He has the habit of saying "OK?" every few seconds, but even good lecturers have their idiosyncrasies. You can see his work clearly; he doesn't stand in front of what he is writing. He erases problems quickly after he finishes, but you can use the "pause" function on the DVD player to take a longer look.
Jason doesn't solve any problems that require finding the unknown sides of a triangle using trigonometry on these DVDs He doesn't cover the Law Of Sines or the Law Of Cosines. He doesn't cover the trigonometric form of complex numbers. When trigonometry or pre-calculus is taught in college, the course usually requires the use of a sophisticated calculator. These DVDs have no examples of using one. (Jason does have another DVD that covers the TI-84, which I have not yet watched.)
Accepting the limited scope of the material, I rate this set of DVDs as four stars out of five to indicate that is an excellent set of marker board lectures on rote problems. Understanding only these problems won't get you an A+. However, if you don't know this material, you'll have a hard time passing.
Synopsis
( I use the customary notation "x^p" to mean "x raised to the p power".)
Disk 1
1) Complex Numbers ( about 40 minutes )
He considers the equation x^2 = -25 and shows how it can be given a solution by defining
i = square roof of (-1).
Thet three types of numbers are: real, imaginary, complex.
He explains a graph of the complex plane. (He doesn't make any further use of the graph.)
He explains how to add complex numbers.
Simplify (3 + 2 i) + (-5 + 4i).
Simplify 7 - (3 - 7i).
He explains the multiplication of complex numbers.
Simplify ( 4 + 3i)(-1 + 2i).
Simplify -9i(4-8i).
Simplify 1/(3 + 2i).
Jason says "Mathematicians don't like to have square roots in the denominator. Well,they don't like to have imaginary numbers in the denominator either." (That's a slander on mathematicians. It's educators and graders who don't like this. They want everyone in class to get the same answer and to perform a little extra work by modifying such fractions. )
He defines the complex conjugate.
Simplify (4 - 3i)/ (2 + 4i).
Solve x^2 - 3x + 10 = 0. He applies the quadratic formula and obtains the complex roots.
2) Exponential Functions (about 21 minutes)
The exponential function is f(x) = a^x. He says the notation "f(x) =" just means "y=".
He explains general form of the graph of f(x) = a^1 for the case a > 1 and the case 0 < a < 1.
He defines a^0 = 1 for "any number 'a'".
He does a "qualitative" graph for each of the following functions:
f(x) = 3^(-x).
f(x) = -2^x. (He should have mentioned that (-2)^x denotes a different function.)
f(x) = 2^(3-x).
f(x) = (2^3)(2^(-x)).
f(x) = e^ x. He introduces the number 'e'. "e is just a very special number."
3) Logarithmic Functions ( about 30 minutes)
"The opposite of an exponential is a logarithm". (Jason never talks about "inverse functions", he likes the term "opposite", but doesn't define it rigorously. He approaches this topic as a kind of "cancellation". If you have a^(log base a of w), the 'a' and the 'log to the base a' will "cancel", leaving you with the 'w'. )
He defines "y = log to the base a of x" to mean "a^y = x".
Find log to the base a of 100.
He lists the significant properties of logarithms without proving them.
If a= e then "log to the base e" is written as "ln"
"log" = "log base 10".
Write the equation 4^3 = 64 as an equation that uses logarithms.
Write the equation 10^(-3) = 0.001 as an equation that uses logarithms.
Write "log to the base 10 of 1000 = 3" as an exponential equation.
Solve: log to the base 3 of (x-4) = 2.
Solve : log to the base 5 of x^2 = -2.
Solve: log to the base 6 of (2x-3) = log base to the base 6 of 12 - log to the base 6 of 3.
Solve: log to the base 10 of x^2 = log to the base 10 of x. He explains why 0 is not a solution
Simplify: log to the base a of ( (x^2)(y)/ (z^3).
He graphs a logarithm function by asserting the fact that it is a reflection of the corresponding exponential graph about the line y = x, which he calls a "45 degree line".
4) Exponential and Log Equations (about 18 minutes)
Solve 10^x = 7.
Solve 3^(4 -x) = 5.
Solve 3^(x+4) = 2^(1-3x). He takes natural logs of both sides.
Solve log x = 1 - log(x-3). ( He eliminates x = -2 as a solution "There is no such thing as the log of a negative number.")
Solve log(5x + 1) = 2 + log(2x -3).
5) Angles ( about 42 minutes )
The best Jason can do to define an angle is "an angle is the measure of how much space there is between two lines". (I forgive him. It's actually very hard to define an angle rigorously. The usual approach in secondary education is to confuse a redundant system of parameterizing angles with the angles themselves. For example, 0 and 2 pi are "coterminal angles" (plural) but they are "really the same angle". Jason uses this approach and students will survive such contradictions. )
He defines "acute angles" and "obtuse angles".
He shows negative and positive angles drawn about the origin of the xy plane.
He shows the angles 180 deg. angle, 270 deg., 360 deg.
He shows angles greater than 360 deg.
He explains that a 450 deg. angle"is really the same measurement as 90 deg".
He introduces radians by declaring that 360 deg = 2 pi radians.
He shows angles of pi/2, pi, 3pi/2.
He draws special angles on an xy graph: 45 deg and multiples are drawn in red, 30 deg and multiples
are drawn in green.
He repeats the drawing, using radians.
He motivates the attention to special angles by saying their trig functions have simple values.
Find two positive angles coinciding with a 120 deg angle.
Find two positive angles and two negative angles that coincide with a 120 deg angle He introduces the terminology "coterminal angles".
Find two positive and two negative angles coterminal with a -30 deg angle.
Find two positive and one negative angle coterminal to an angle of (5 pi)/6.
Convert 150 degrees to radians. He explains the cancellation of units when using a conversion factor.
Convert -60 degrees to radians.
Convert 225 degrees to radians.
Convert (2 pi)/3 radians to degrees.
Convert (11 pi)/6 radians to degrees.
Disk 2
6) Finding Trig Functions Using Triangles (about 27 minutes)
He defines sine, cosine as tangent as ratios of sides in a right triangle.
He defines cotangent, secant and cosecant as reciprocals of the previous three functions.
He hints about the relation of sine,cosine and tangent to coordinates in the xy-plane.
Find all the trigonometric functions of the angle theta that is adjacent to a side of length 3 in a "3,4,5 " right triangle.
He mentions that tan(theta) = sin(theta)/cos(theta).
Find all the trigonometric function of the angle theta that is opposite to a side of length 2 in a right triangle whose hypotenuse has length 5.
7) Finding Trig Functions Using The Unit Circle (about 53 minutes)
He writes a table that shows the values of the sine,cosine and tangent of "special angles", pi/6, pi/4, pi/3. He says to memorize the table. (Jason doesn't derive these values. The usual way to do that would be to analyze an equilateral triangle with an altitude drawn in it and a square with a diagonal drawn in it.) He draws the unit circle in Cartesian coordinates.
He explains that the sine and cosine of an angle are coordinates of a point on the unit circle.
Find sin( (5 pi)/6 ).
Find cos( (5 pi)/6).
Find sin ( (2 pi)/3).
Find sin ( (4pi)/3).
Find cos ( (5pi)/6).
Find cos ( (7pi)/6).
Find sin(0).
Find sin ( pi/2).
Find cos (pi/2).
Find sin ( (3pi)/2).
Find cos ( (3pi)/2).
Find sin (pi).
Find cos(pi).
Find sin( -pi).
Find cos(-pi).
Find tan( (-5 pi)/4).
Find tan ( (-3 pi)/4).
He explains the notation "sin^-1".
Find inverse sin(1/2) ).
Find inverse sin ( (square root of 3)/2).
Find inverse cos( (- square root of 2)/2).
8) Graphing Trig Functions (about 51 minutes)
He explains how to see how the sine and cosine of an angles change as the angle changes by visualizing this on a unit circle.
Graph y = sin(theta).
He defines the amplitude and period of the wave.
Graph y = cos(theta).
Graph y = 2 sin(theta).
Graph y = sin(2 theta). (He writes the problem as "y = sin(2x)".)
Graph y = sec(theta).
Graph y = csc(theta).
Graph y = tan(theta).
9) Trig Identities (about 39 minutes)
He writes the trigonometric identities cot = 1/tan, sec = 1/cos, csc = 1/sin.
He writes the identity sin^2(theta) + cos^2(theta) = 1.
He explains the notation "sin^2".
He explains the identity using the unit circle and the Pythagorean theorm.
He writes the identity 1 + tan^2(theta) = sec^2(theta).
He writes the identity 1 + cot^2(theta) = csc^2(theta).
Verify the identity: cos(theta) sec(theta) = 1.
Verify the identity: sin(theta) sec(theta) = tan(theta).
Verify the identity: (1 + cos(theta))(1 - cos(theta)) = sin^2(theta).
Verify the identity: sin(theta)/csc(theta) + cos(theta)/sec(theta) = 1.
Verify the identity: sec(theta) - cos(theta) = tan(theta) sin(theta).
Verify the identity (sec^2(theta) - 1)/ sec^2(theta) = sin^2(theta).
Verify the identity sec^2(theta) csc^(theta) = sec^2(theta) + csc^2(theta).
Verify the identity (cot(theta) -1)/(1 - tan(theta)) = cot(theta).
Verify the identity: (sin(theta) + cos(theta))/(tan^2(theta) -1) = cos^2(theta)/ (sin(theta) - cos(theta)).
(Jason uses the careless approach to verifying identities that is traditional in secondary education. For example, in one problem he multiplies both sides of "the equation" by cos(theta) without worrying about whether cos(theta) might be zero. Coach Glonther wouldn't count this wrong.)
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