Thursday, February 19, 2015

Sections 14-3 and 14-4: The Sine, Cosine, and Tangent Ratios (Day 115)

Today is Chinese New Year, one of the most celebrated holidays in the world. So now is a good question to ask, how is Chinese New Year calculated?

Recall what I wrote at the Western New Year regarding calendars. We can describe them in terms of which boxes from that calendar checklist would be checked. For the Chinese Calendar, we'd have:

(x) the lunar month cannot be evenly divided into solar days
(x) the solar year cannot be evenly divided into lunar months
(x) having months of different lengths is irritating
(x) having months which vary in length from year to year is maddening

The Chinese Calendar is a lunisolar calendar -- indeed Chinese New Year is often called "Lunar New Year," in both China and other nearby Asian nations that celebrate it.

The following link describes the Chinese Calendar in more detail:

Here are a few comments about this calendar, as mentioned at the link:

The Chinese Calendar assumes a prime meridian of 120 degrees East (120°E). This means that a day (or rather, a nychthemeron, a day and a night) is taken to run from midnight Beijing standard time (BST = CCT = GMT+8) to the next midnight BST.

Oops, I forgot to check one of the boxes:

(x) Greenwich is not unambiguously inferior to any other possible prime meridian

Of course, the Chinese calendar has existed for thousands of years, before anyone from China had ever heard of Greenwich, so to them Greenwich was inferior to a prime meridian through China.

As we read the link above, we see that not only New Year, but every month is supposed to begin at an astronomical dark (or "new") moon, no matter what. This means that not only do months have different lengths, but the same month has can have different lengths in different years (although it's always either 29 or 30 days). Here astronomical exactness has priority over simplicity of month or year lengths, as there is always a trade-off between the two.

Why is Chinese New Year always in January or February? Let's see:

New Year's Day in the Gregorian Calendar always occurs about a week after the northern winter solstice, whereas on average New Year's Day in the Chinese Calendar occurs approximately midway between that solstice and the northern vernal equinox.

The day that is halfway between the winter solstice and spring equinox is called Lichun in Chinese -- this means the same thing as Imbolc to the pagan Wiccans. (Westen Christians often refer to this as Candlemas or Groundhog Day.) There is often a debate as to when the seasons begin. In China, the seasons run as follows:

Spring: February through April
Summer: May through July
Fall: August through October
Winter: November through January

The winter solstice is the darkest day of the year, while the coldest day of the year often doesn't occur until almost Lichun/Imbolc. Likewise, the lightest day of the year is the summer solstice, but the hottest day isn't until July or even August. The temperature trails the light by about a month or two, especially near the oceans due to the high specific heat of water. This means that using the Chinese definition of season, the hottest season of the year is often the fall and the coldest is the spring.

Nonetheless, to the Chinese, Lichun is the luckiest day of the year, and so a year containing two Lichuns -- a year containing a Leap Month, a 13th lunar month -- is also lucky. The year that just ended, the Year of the Horse, was a lucky year containing two Lichuns, while the new year, the Year of the Sheep, is an unlucky year with no Lichun.

Here's another webpage, from Singapore, about the Mathematics of the Chinese Calendar. It refers to the current unlucky year as a "Double Blind Year":

Section 13-3 of the U of Chicago text is on the tangent ratio, and Section 13-4 of the U of Chicago text is on the sine and cosine ratios. I have decided to combine all three trig ratios into one lesson.

David Joyce was not too thrilled to have trig in the geometry course. He wrote:

Chapter 11 [of the Prentice-Hall text -- dw] covers right-triangle trigonometry. It's hard to see how there's any time left for trigonometry in a course on geometry, but at least it should be possible to prove the basic facts of trigonometry once the theory of similar triangles is done. The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?) The one theorem of the chapter (area of triangle = 1/2 bc sin A) is given for acute triangles.
As the trig functions for obtuse angles aren't covered, and applications of trig to non-right triangles aren't mentioned, it would probably be better to remove this chapter entirely.

Yet most geometry books include trig because most state standards require it. And this most certainly includes the Common Core Standards:

Define trigonometric ratios and solve problems involving right triangles

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

And all three of these standards appear in this lesson.

Some people may wonder, why do we use the same name "tangent" to refer to the "tangent" of a circle and the "tangent" in trigonometry? A Michigan math teacher, Mike Shelly, discusses the reasons at the following link:

On the other hand, the reason that "sine" and "cosine" have the same name is less of a mystery. In fact, the U of Chicago tells us that "cosine" actually means complement's sine -- since the cosine of an angle is the sine of its complement. This is Common Core Standard C.7 above.

OK, let me post the worksheet, and wish everyone a very happy Chinese New Year!

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