Quandaries
and Queries 

Hi, my name is Amanda, and I'm going into my senior year of high school. I will be taking AP calculus, and my teacher gave us some homework over the summer. However, there are two things that I do not understand how to do. The first is, she wants us to be able to generate a unit circle by hand using 30, 60 and 90 degree triangles. I have used the unit circle in trigonometry, however I was never taught how to draw it. Secondly, I need to know how to do natural logarithms without a calculator. I was not taught how to do this, and the worksheet I was given only showed me how to complete them using a calculator. Thank you so much for any help that you might be able to offer Amanda 

Hi Amanda, This looks like a
very tough class. I was taught how to draw a circle using compasses,
pennies or glasses, but never with triangles. Also, I know how to compute
the natural logarithm using tables, a slide rule or a calculator, but
without these tools, the best I can do is approximate them. (And even
Napier http://turnbull.mcs.stand.ac.uk/history/Mathematicians/Napier.html, To approximate natural logarithms, you can make a small table as follows: the base e is about 2.7, so that ln(2.7) is approximately1. Then, e e is approximately 7.3, so that ln(7.3) is approximately2. Then, e e e is approximately 19.7, so that ln(19.7) is approximately 3, and so on. In this way, you make
up a list 2.7, 7.3, 19.7, 53.2, ... of numbers whose There is another way to approximate natural logarithms if you are allowed to use one of these solar powered calculators with a square root button but no ln button:
On my calculator, I get 2.9849... when I perform these operations. This is quite good since it is supposed to be an approximation of ln(19.7) which is about 3. You can try it with 2.7 and 7.3 to convince yourself that it works well enough. Then one day in your second or third calculus course you will learn why this works. Claude 

