Thursday, February 7, 2008

Exact Values In The Unit Circle

Hello everyone, I'm Nelsa, today's scribe. I'm so sorry for the late post, Thursday is a really busy day for me.

Mr. Kuropatwa began the class by asking everyone if we have read the 'Digital Ethics' post and taking everyone's oath (by shaking our hands) that we'll abide by those rules. He also explained 'Blogging On Blogging' - which you can learn more about by reading that post - and how to properly label your posts. But of course, this is a math class, and that's probably what you all are waiting for. Most of the things we studied today, reviewed a lot of what we learned yesterday, so I'm sorry if it seems like I'm repeating the things Ben has detailed on the last scribe post.

The first slide required us to find which quadrant P(5) can be found in. Most of us were more than slightly confused because of the lack of a y-coordinate, but Mr. K simply told us to think in 'radians'. With this in mind, we reviewed what we already knew, which was, 180°=π and 360°=2π. π is approximately 3, which means that 2π would be approximately 6. In order to get 5, we subtract one from six, and whenever we subtract, we move in a clockwise direction, which would mean that P(5) is in quadrant IV.

We were also asked to find which quadrant contained sinθ <> 0. To do this, Mr. K told us to forget all about 'CAST', which I'm sure we all learned last year, and instead, understand what's going on. Cosine, as we know from 'SOHCAHTOA', is the x-axis or coordinate, and sine, is the y-axis or coordinate. Keeping this in mind, we know that in quadrant I, both cosine and sine is positive, as both the x-axis and the y-axis is positive. This also means that tangent is positive, because when you divide two positives (which is really what tangent is, sine/cosine), it's a positive. In the second quadrant, cosine is negative, because it's before the zero, but sine is still positive, which means that tangent is negative. In the third quadrant, both cosine and sine is negative, as both are before the zero, which means tangent is positive. Finally, in the fourth quadrant, cosine is positive, and sine is negative, making tangent negative as well. Using this information, we were able to answer the first question on the second slide.

The equation of a unit circle is x(squared) + y(squared) = 1. The third slide asked us to determine whether the point (1/root of 5, 2/root of five) is on the circumference of the unit circle. So using the above formula, we figured out that the point (1/root of 5, 2/root of five) is on the circumference of the unit circle. But a simpler way to find the answer is to understand, again, that cosine is the x-coordinate, and sine is the y-coordinate. The radius of a unit circle is 1 (hence 'unit'), so the cosine squared, plus the sine squared should equal to 1, which is the hypotenuse.

Mr. K then went on to talk about the person, who died because of one of the triangles included in your basic geometry set. I forgot the person's name, but basically, a group of people who call themselves the Pythagoreans believed that the world was made up of (only) rational numbers. One of the triangles in a geometry set, besides having a 90° angle, also have 45° angles. The hypotenuse of this triangle, is the root of 2, which is an in irrational number. The Pythagoreans were aghast, and kept this fact hidden from the rest of the world, as it would ruin everything that they believed in. But one person from the inner circle (who were the only people who knew) told everyone else, and, to make a long story short, was thrown off a cliff.

So that's basically what we learned in class today. Again, I'm so, so sorry for the late post. The next scribe is twenty-seven hundred, ninety-eight, twenty, ELEVEN, nine. =)

1 comment:

Lani said...

Hi Nelsa,

Thanks for your explanations!

What a great story! Is this who you were referring to?
Hippasus

Best,
Lani