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Following on this theme we get as our radius, assuming 360 as a circumference in both degrees and distance, a distance of 114.59155902616464175359630962821 and divide by 2 gives us: 57.295779513082320876798154814105

Get out any calculator, the one with Windows will be fine and find the tan of 89.9999999

There is a good reason for this. The natural unit for the angular measure in circles is the radian. 1 radian is defined as the angle subtended by an arc of length equal to the radius of teh circle. Since a circle consistes of 360 degrees, and teh length of the circumfrence is 2PIr, then there 2PI radians in a circle. Thus one radian is equal to 360/2Pi = 57.295977951 degrees. This is similar to what Don has done (more on this later).

Now lets look to see how a calculator determines the value of tan(89.99999999). Some may be surprised to learn that the calculator does not work in degrees, but rather in radians, but then converts to degrees. There is even a radian button on calculators to put it back into natural units. And again I emphasis that the natural unit of the circle is the radian, not teh degree which is merely a rescaling by man.

The tan of an angle is calculated from the ratio of the sine to cosine of an angle i.e. tan = sin/cos. Note though that sin(q) = cos (90-q) and vice versa. The calculator has real trouble calculating values that diverge rapidly to infinity. take the tan of 90 degrees and you receive an error, because tan 90 = infinity. this is because the sin of 90 degrees = 1 but cos90 = 0, and 1/0 = infinity.

So to help illustrate how a calculator determines the value of 89.9999999 we can do the calulation ourselves without using the calculator.

we know that tan(q) = cos(90-q)/sin(90-q) = cos(0.00000001)/sin(0.00000001). These are in degrees so lets convert to radians using the conversion factor that 1 degree = 1/57.295977951 = 0.017453293 (this is the only part you may need a calculator for, but it can be done with pen and paper)

tan(q) = cos(1.745329252e-10)/sin(1.745329252e-10)

So lets use the approximation method of angles when expressed in radians. The cosine of a very small angle in radians is approximately equal to 1 (the smaller the angle the more accurate the approximation). The sine of very small angle in radians is equal to the angle itself (again the smaller the angle, the more accurate the approximation). So we have here then that

Tan(q) = 1/1.745329252e-10

This is the same as 1/[(1/57.295977951)*1e-8] which is the same as 5.72977951e9.

So Tan(89.9999999) = 5.72977951e9.

In fact if you increase the number of 9's after the decimal, you are effectively making the approximation more accurate in the small angle regime, and increase the size of the answer. For every 9 you have after it, makes the answer approximately 10 times bigger. the reason why you get the values that you do is because of the calculator using radians to calculate instead of degrees, and because it cannot operate with infinite precision.

A calculator cannot calculate with infinite precision and calculates using radians, that is why when you put in the above values you get an answer that seems to mimic a circle of length 360 units. There really is no inherent mystery to the values. All don has done is scaled all measures of his circle by a factor of 57.295977951..... transforming a circle of radius 1 and circumference of 2Pi into one of radius 57.295977951... and circumference of 360.

Jonny

The path to good scholarship is paved with imagined patterns. - David M Raup



Edited 1 time(s). Last edit at 05/25/2010 10:29AM by JonnyMcA.