Omg Sin And Cosine Have Been Eliminated

Here we go:

quadrance = (distance)2

So the quadrance Q (A1, A2) between the points is
Q (A1, A2) ≡ (x2 − x1)2+(y2 − y1)

Spread is slightly harder to derive:

spread = ( sin (angle))2

s (l1, l2) =
Q (B, C)
----------
Q (A, B)
=
(a1*b2 − a2*b1)2
-------------------
(a12+ b21) (a22+ b22)

 

If you are interested in the math, it's all in the chapter 1 preview pdf on his site.

BTW the equivalency of spread is not so important as the way it is actually calculated (the ratio of quadrances) and the way you can use it to solve trig problems (without ever touching a sine).

 

This might sound strange, but Sin translated in Italian (="seno") means breast.
Imagine poor maths teachers introducing their pupils to "This new thing were starting today called Breast!" and getting histerical laughs every single time... I don't think any class was ever immune from laughs even long after they were introduced to Trigonometry (which i loved BTW).

 

I thought it was "sine" in English, not "sin". Which looks rather humorous like "seno" in Italian, by the way.

 

Yeah, in English it's quite ominous... i mean... "your Sin will make you fail the finals!"

In Italian it's just funny, "Your breast is wrong"... "Your Breast is too big, it should be this much"

 

to me it just looks like a different way of explaining dot products. (without actually explaining dot products)

 

I was going to say that too Graham

 

Yeah, well, there's no way in hell you can get rid of sin/cos functions (and definitely not complex exponentials: e^(ix) = cos(x) + i*sin(x), where i = sqrt(-1), a formulation of sin/cos which is more compact and easier to work with for many operations), as these functions are the solutions to many very useful differential equations. They are used routinely for signal analysis (fourier transforms) and a great number of physical phenomena (from ocean waves to perturbations on the cosmic microwave background).

So it would be a horrible mistake to try to teach students this sort of trigonometry instead of the standard trig in class. It's pretty important to be familiar with sines and cosines when entering differential equations and physics classes. But it may have some applications for computational purposes.

 

This is just a trivial algebraic manipulation of the basic trig formulas.

First, square distances so that you don't need to take a square root to use the Pythagorean theorem. Whoopdedo.

Next, since sin(angle) = y/r, square it to get sin^2(angle) = y^2/r^2. Now, tell your students to measure y^2 and r^2 with their new quadrance rulers, instead of using a protractor. Call Sine Squared "Spread"

Step 3, take all of the basic trig formulas and recast them as quadrance/spread.

Yawn!

Revolutionary my ass. Sure, it makes some formulas simpler by "eliminating" square root symbols and "sine", it is bound to make other formulas more complicated.

Some formulas are naturally more elegant in polar coordinates for example, and some are naturally more succint in cartesian.

This "New Trig" will succeed in just one thing: dumbing down the curriculum.

Using calculators is not the problem that makes trig difficult, and the use of irrationals is irrelevent.

 

It is. It just gets abbreviated on calculators etc.

 

Not just on calculators. It's pretty standard to write the function as sin(x) when writing equations on paper.

 

Point is, why people here abbreviate sine to sin but not cosine to cos? The topic itself reads "...Sin and Cosine...". Wouldn't it be more handy to abbreviate Cosine instead?

 

Well, in equations people do abbreviate both. Anybody that writes, "sin and cosine," is being really inconsistent. Maybe some people just don't realize that sin is the abbreviated form of sine (whereas cosine is more obvious, since it's almost never pronounced anything like "cos").

 

Errrr it is, actually. Generally it sounds 1/2 way between "coz" and "coss".

 

Hrm, well, I guess I talk math with different people than you do, then. But regardless, I'm sure most people have heard at least one person use the term Cosine instead of cos.

 

Using a calculator when there is no real need to shows you are doing something wrong IMO.

 

Therefore, "sin" is pronounced like "sin" and not like "sine". Weird...

 

Who said that?

 

Well, there's no real need to even calculate final numbers. They could leave everything as radicals and trig functions and leave it at that.

What you want the student to have is correct REASONING, the fact that he can then do basic arithmetic by adding the numbers without a calculator is irrelevent. By the time a student learns trig, they shouldn't have to do all numerical calculations by hand. As a teacher of trig, I don't want to evaluate whether the student can do long division, I want to evaluate whether they can do the proofs or solve the problems from a logical and analytical point of view.


All this "New Math" really does ultimately is allow the student to leave his final answer in the form of distance squared, and the teacher will accept it, rather than forcing him to take a square root and leave the answer in the form of a distance.

I say, just allow the student to leave the answer in the form of a formula, which if evaluated, would yield the correct result. The TA grading the papers can evaluate the formula if they wish.

Realistically, they should even allow students to write programs in class if they want to calculate answers. Let the students use whatever tools for the job they need to solve the problem. Ultimately, it's about solving problems, not how good one is at arithmetic.

 

Yes, and problem solving is something most students of today are woefully-inadequate at doing. Analytical problem solving is the essence of physics, and the vast majority of the lower-division physics classes I've TA'd for (both calculus-based and not) had very, very few students with any problem solving skills.

Edit: Note that this is at a pretty good university (University of California Davis).