Thursday, May 4, 2017

Lesson 14-7: Adding Vectors Using Trigonometry (Day 147)

This is what Theoni Pappas writes on page 124 of her Magic of Mathematics:

"The profound study of nature is the most fertile source of mathematical discoveries. -- Joseph Fourier"

This is the first page of a new section, on hexagon. Pappas writes that hexagons appear in nature -- and it's not just in honeycombs. Indeed, on this page is a photo of hexagons naturally appearing in rock, taken at Mammoth Lakes, which is right here in California. In this section, Pappas will explain why hexagons are so special.

By the way, Joseph Fourier was an early 19th century French mathematician. He was famous for coming up with Fourier series -- which are a bit like Taylor series in Calculus, except that a function is written as a sum of sine waves rather than polynomials. (My AP Calc teacher, in her zeal to find a modeling problem, once accidentally assigned us a Fourier series!)

Fourier series and sine waves remind me of trig -- and speaking of which...

This is what I wrote last year about today's lesson:

Lesson 14-7 of the U of Chicago text is on adding vectors using trigonometry -- and we can't skip it because it appears in the following Common Core standard:

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Notice that we are essentially converting the vectors from polar to rectangular form, adding their components, and then converting the sum back to polar form. And all of this is done without the students' even knowing what polar coordinates are.

In an actual trig course (as part of precalculus), we find out that we can avoid thinking about angles larger than 90 degrees by considering the reference angle -- the angle formed by the x-axis and the terminal side of the original angle. We see that even though the U of Chicago's Geometry text doesn't teach reference angles, the angles shown in the text are always formed using the x-axis -- that is, the west-east axis -- and never using the y-axis. So the U of Chicago, while not explicitly teaching reference angles, clearly has these reference angles in mind when writing this chapter.

The text states that one of the two components is found using the sine of the angle, while the other is using the cosine. But because of the way the angles are drawn, the horizontal component will always use the cosine and the vertical component will always use the sine -- just as they would be for polar coordinates in a trig class.

To convert from rectangular back to polar coordinates, we use the distance formula and the inverse tangent function. This is the only time that an inverse trigonometric function appears in the U of Chicago text -- although many other geometry texts discuss inverse trig in more detail.

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