[InterceptPoint]

Take a look at Chapter 1 of the book. What I see is a complex replacement of sin/cos/tan with what amounts to the Pythagorean Theorem. You have to give up the concept of angle and distance and replace it with "quadrance" (distance squared) and "spread" (don't ask) and you still (obviously) have to be able to do square roots to get answers. So he makes you give up the intuitive idea of distance and angle just so you don't have to push the SIN, COS or TAN buttons on your calculator. Instead you have to push the SQR button several times.

This is progress?

[Flashlight]

ok now I get it.

now I'm gonna write a book that replaces sin(x) with exp(ix)-exp(-ix)/2i, cos(x) with exp(ix)+exp(-ix)/2, and tan with sin/cos. exp(x) is easily calculated using the Taylor series 1+x+x**2/2!+x**3/3!...

but seriously folks, somebody should shoot that stupid sohcahtoa stuff.

sin, cos, and tan aren't three functions, they are essentially only two - sin and cos are the same function, they just start at a different place (it's like making a distinction between the split end and the flanker in football - yes they have to line up correctly, but essentially they're the same thing: they're both wide recievers).

can't explain everything here of course, but it's easy to intuitively grasp the concepts of sin and cosine by knowing that cosine is the "projection" onto a surface and sin is the "height" (multiplied by the length of the segment that's being "projected") - this is a whole lot easier to see and understand than to explain in writing here.

the point is, to really "understand" this stuff, the student should just be able to look (or visualize) at a problem like "if a stick of length 4 is sticking out of the ground at a 34 degree angle, what's the distance from the top of the stick to the ground?" and just "know" that, well sin is the height and the stick is length 4, so the answer is 4*sin(34 degrees).
anyone who resorts to formulas like hypotenuse/ajacent or whatever may end up with the correct answer but they're lacking the essential intuitive knowledge that will serve them as they progress in trig and calc.

[Doctor Stochastic]

This is progress?

On a cheap calculator, it is.

[Right Wing Professor]

Another issue is that square roots are not uniquely defined. This means his spread function is defined only over a and of 0 to 90 degrees, rather than -90 and 90. I can uniquely define a point in a 2D frame as r sin theta. Defining it as r^2 * sin^2 theta introduces a four-fold redundancy.

[Excuse_My_Bellicosity]

Okay, I read Chapter 1. I'll take an angle and a distance over quadrance and spread any time, and I swear I am trying to read this all with an open mind. The author claims that he's simplifying trigonometry. So far, I'm not seeing the simplification here.

My problems with what I'm seeing so far are:

• Regarding the distance between 2 points, he states "quadrance is the more fundamental quantity, since it does not involve the square root function." That is not a true statement. Try explaining both concepts to a 5 year old and then tell me which is simpler and more fundamental. When riding a bike down the old dirt road, determining gas mileage, or doing anything else that involves a one-dimensional distance, I care about the distance a hell of a lot more than the "quadrance". When Dr. Wildberger takes away the square root symbol from a Cartesian distance calculation and calls it something else ("quadrance"), that doesn't make the problem simpler, it only means that he has to incorporate the square root in subsequent calculations (unless it simplifies out, which doesn't happen in Dr. Wilberger's calculations; at the end of his example, he performs D = SquareRoot(Q)). I don't view this as simplification.
• The equation: s(l₁,l₂) = Q(B,C) / Q(A,B) = (a₁b₂-a₂b₁)² / (a₁²+b₁²)(a₂²+b₂²) is a simpler concept than an angle????
• "The problem is that defining an angle correctly requires calculus." That's a true statement, but really defining Dr. Wildberger's coordinate systems and quantities also requires calculus, a fact that he glosses over. When I took AP calculus in high school college calculus courses, I had to use calculus to really prove that the circumference of a circle is C = pi * D = 2 * pi * R and various other fundamental mathematical principles but that does not mean that I had to do so before learning basic geometry, trigonometry, and analytical geometry. The good doctor is purposely introducing a moot point in an attempt to describe why his system is supposedly better.
• "Without tables, a calculator or calculus, a student has difficulty in answering this question, because the usual definition of an angle (page 11) is not precise enough to show how to calculate it. But how can one claim understanding of a mathematical concept without being able to compute it in simple situations?" Uh huh. But a student is supposed to able to compute s(l₁,l₂) = Q(B,C) / Q(A,B) = (a₁b₂-a₂b₁)² / (a₁²+b₁²)(a₂²+b₂²) without tables or a calculator?
• "If the notion of an angle cannot be made completely clear from the beginning, it cannot be fundamental." That is not a true statement. Mathematical proof can be difficult but an angle is easily visualized. It's not a difficult concept. And the author does not show that his concept of "spread" is any more simple or fundamental than the angle.
• Section 1.5 shows a comparison of solutions for a problem using conventional trigonometry and Dr. Wildberger's method. Classically, I can solve this using the Law of Sines such that d = (5 * sin 41.4096 deg) / sin 93.5904 deg = 3. 3137. Using the other approach, I have to find the root of the equation: (7/16 + 1/2 + r)² = 2(49/256 + 1/4 +r²) + 4 * 7/16 * 1/2 * r. Call me crazy, I'd rather punch up a couple of sines than find that root!
• "Clearly the solution using rational trigonometry is more accurate." You might want to expand on that thought, doc. You haven't shown me that it is. In a mathematical expression, writing the solution with the square root of 7 intact is more accurate than the decimal expression of the anwswer, but I can make the classical answer the same accuracy as Dr. Wilberger's by not simplifying for the sine or cosine. And it should be noted that the numerical methods that a calculator uses to derive a decimal for the square root of 7 is no more accurate than it is for sin, cos, or tan of angle XX.XXXX....
• "Defining arc lengths of curves other than line segments is quite sophisticated, even for arcs of circles. So it does not make mathematical sense to treat circles on a par with lines, or to attempt to use circles to define the basic measurement between lines." A circular arc length, L = (angle in degrees) / 360 * pi * diameter, is a difficult concept?

I'd like to critique the rest of his book but I'm not about to fork over the $80 to do it.

[Allan]

There is a humorous book called Mathematics Made Difficult
where basic topics in mathematics
(including trigonometry)
are made so complicated they are impossible to understand.

I think the present author could contribute a chapter to the next edition.