How to derive the Law of Sines, and the Law of Cosines

When I was teaching the Law of Sines and the Law of Cosines I decided to challenge myself by actually deriving each equation. The Law of Sines was pretty easy, but the Law of Cosines took me 3 tries before I was able to get it. Here they are.

Walk or Run in the Rain

For a couple years I have wondered about whether you get wetter by walking or running in the rain. I know the Mythbusters have attacked this question, but don't remember their conclusion. I decided to try to tackle the problem mathematically, and while I made some assumptions, I think my conclusion makes sense.

How Many Different Calendars Are There?

While talking with my friend Cody about the possibility of reusing calendars every few years due to them repeating, we wondered if every possible calendar could occur, and how many years one would have to wait to have all the possible calendars. When I got home, I made a spreadsheet that identified the pattern.

It turns out that if we started saving calendars this year, we would have all possible calendars by 2036, and that some of the calendars from common years would begin repeating in 2019, so from 2019-2036 we would only need to save the calendars from leap years.

This pattern would continue until 2099. The rules for leap years would make 2100 NOT a leap year and the pattern would start over again in 2101. The rule for leap years is that every year that is divisible by four, except for multiples of 100 that are not divisible by 400 ARE leap years.

Classifying Quadrilaterals

Julie was working on how to classify quadrilaterals for her class, so I made up this chart that kind of breaks it down. I thought it turned out pretty good, and it gave me a chance to use my Kevin font. The one thing I didn't include was kites which have two sets of congruent sides that are adjacent to one another.

Big Numbers

Most people don't know a lot of large numbers. I myself do not know a lot of large numbers, but I know more than most people, and I am going to share some with anyone who wants to read about them. Most people know thousands, then millions, and billions. A large number of people probably even know that after billions comes trillions. But what comes next? Is it Gazillions? Bazillions? Nope, quadrillions. Here is the breakdown...not all of them, but enough to sound smart.

1,000,000,000 (1 x 10^9) One Trillion

1,000,000,000,000 (1 x 10^12) One Quadrillion

1,000,000,000,000,000 (1 x 10^15) One Quintillion

1,000,000,000,000,000,000 (1 x 10^18) One Sextillion

1,000,000,000,000,000,000,000 (1 x 10^21) One Septillion

1 x 10^24 One Octillion

1 x 10^27 One Nonillion

1 x 10^30 One Decillion

skip ahead

1 x 10^99 One duotrigintillion

1 x 10^100 (a 1 followed by 100 zeroes) Commonly called a googol (not google) it could also be called ten duotrigintillion.

1 x 10^(10^100) would be a googolplex. That would be a 1 followed by a googol of zeroes.