Friday, February 20, 2015

Activity: The Sine, Cosine, and Tangent Ratios (Day 116)

I'm still thinking about the Chinese New Year. As it turns out, 2015 is an exceptional year in the calculation of the date of the new year. As I mentioned yesterday, Chinese New Year falls near Lichun (or Imbolc), the midpoint between the winter solstice and the spring equinox, and that it falls at a dark (or new) moon. Indeed, Chinese New Year is almost always the dark moon that's the closest to Lichun. But one exception is 2015. According to the link I mentioned yesterday, this is because Lichun falls almost exactly halfway between two dark moons (i.e., at a full moon). The rule is quite complicated and is only explained fully at the link from yesterday.

The following link mentions that the past Year of the Horse contained a Leap Month -- that is a 13th month, since the solar year can't be divided evenly into lunar months. Unlike our solar calendar, where the Leap Day is always February 29th, or even the Hebrew lunisolar calendar, where the Leap Month is always in winter (the month "Adar" is repeated), in the Chinese calendar, any month can be the repeated month. According to the following link, the ninth month was the repeated month:

http://ssquah.blogspot.com/2014/11/li-chun-2015.html

Finally, here's a link to the Archetypes Calendar, a calendar with 10 days per week. It is a rule-based calendar, rather than astronomical, but it turns out to match the Chinese calendar almost perfectly.

http://www.hermetic.ch/cal_stud/arch_cal/arch_cal.htm

I could go on and on about the mathematics of the Chinese calendar, but that's enough for now. This is a geometry blog, not a Chinese calendar blog!

Here are the activities that I've decided upon for this activity day. All of them are are discovery activities found in the Exploration Questions in the U of Chicago.

(From Section 14-3: The Tangent Ratio)
Choose three angle measures (other than 90) whose sum is 180. (For example, you could choose 25, 97, and 58.)
a. Using a calculator, find the sum of the tangents of the numbers you have chosen.
b. Calculate the product of the tangents of the numbers you have chosen.
c. Repeat parts a and b with a different three numbers.
d. Make a conjecture based on what you find.

The goal of this activity is for the students to discover the surprising identity: if A, B and C are three angles whose measures add up to 180, then tan A + tan B + tan C = tan A tan B tan C. A full proof of this fact requires the addition formula for tangent, which one doesn't learn until a full trigonometry course later on. Of course, none of the angles can be 90, since the tangent of 90 degrees is undefined.

(From Section 14-4: The Sine and Cosine Ratios)
a. Fill in this table of values of the sine and cosine using your calculator.
(In the given table, all multiples of 5 degrees, from 0 to 90 degrees, are given.)
b. For which values of x does (sin x)^2 + (cos x)^2 = 1?

Obviously, the goal is for the students to discover the Pythagorean identity.

(From Section 14-5: Vectors)
a. Find out how long it takes by airplane to go from a nearby airport on a nonstop flight to some other location, and how long the return flight takes.
b. Allowing some time for takeoff and landing (from 5 minutes at a smaller airport to 20 minutes at the largest airports), about how fast does the schedule assume the plane can travel? Is there any assumption about wind?

As it turns out, a round trip flight can be shorter going one way than returning. Flying from California to New York is about 30 minutes shorter than flying from New York back to California. Of course there's an assumption about wind -- one is flying with the wind when going east and against the wind when flying west. This is what the students are to discover, and this is an introduction to the concept of vector.

This question requires research, but that should be easier now in the age of the Internet than back when the U of Chicago text was written. All three problems require calculators -- obviously for the trig questions, and likely for the last question as well, since students will have to divide numbers that almost certainly won't come out even to calculate the speed.