Sine numbers in Charts for AnglesDate: 12/22/97 at 21:21:15 From: Paul Oswalt Subject: Where do the sine numbers come from in the charts for angles? Hi. I am a 39-yr.-old toolmaker. I would like to know where the numbers in sine charts or on a calculator come from. I probably learned this in school, but that was a while back. I use these numbers every day, and was curious how they are derived. I have looked in a couple of books I have, but no luck. I am guessing Pi has something to do with it, but in trying I couldn't come up with the right answers. If you can help, thanks. Paul Oswalt Date: 01/05/98 at 09:24:54 From: Doctor Bruce Subject: Re: Where do the sine numbers come from in the charts for angles? Hello Paul, There is a formula to calculate the sine of an angle. The formula is what we call a series: it is, formally, the sum of infinitely many numbers. Fortunately, the numbers dwindle in size very rapidly, so we can use just the first few terms of the infinite series as a good approximation. Here are the first 5 terms of the series for sine. sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ... I'm sure you can see the pattern. What may look strange are the numbers 3!, 5!, 7!, 9! in the denominators. These are equal to 6, 120, 5040, and 362880, respectively. We calculate n! (read "n factorial") by multiplying all the whole numbers from 1 up to n (including n). So these numbers get large very fast. 100! is a number with 158 digits, for example. You guessed that Pi has something to do with all this, and you are right. It turns out that if we substitute the value Pi/2 (about 1.57) for x in the formula given above, the value will come out very close to 1. If we include more terms we will come even closer to 1. In a certain sense, the sum of "all" the infinitely many terms in the series is exactly 1. So we find sin(Pi/2) = 1. But what kind of "angle" is Pi/2? Most real-world applications of trigonometry use the standard system of measuring angles in degrees. But in order to use this formula for sine, we have to use a different unit of measure. It is called a radian. One radian equals approximately 57 degrees, 17 minutes, 45 seconds. Pi radians equals exactly 180 degrees. So, Pi/2 radians (the number I substituted above) equals 90 degrees. I am sure you know that the sine of 90 degrees is 1. Hope this helps, -Doctor Bruce, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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