minutes and degrees: statute miles in nautical miles

[messa]

Question: If angle ACB has measure 1', then the distance between A and B is a nautical mile. Approximate the number of land (statute) miles in a nautical mile. (Line CB) Radius = 4000 miles

I googled "how many statute miles in a nautical mile". The internet says 1.15 miles is in a nautical mile. But what I did to find the answer was s=(radius)(theta) so I did (4000)(1/60) this equals 66.67 miles. This is quite different from the 1.15 miles the internet says!

Could you please help me work through this a different way than "the computer says so".

 

[skeeter]

\(\displaystyle \L s = r\theta\) only works if \(\displaystyle \L \theta\) is in radians, not degrees.

1/60 of a degree = pi/(60*180) radians

you should get closer with this ...

\(\displaystyle \L s = 4000 \left(\frac{\pi}{60*180}\right)\)

 

[galactus]

The formula you're using. The well-known \(\displaystyle s=r{\theta}\), only works in radians. Multiply by \(\displaystyle \L\\\frac{\pi}{180}\)

\(\displaystyle \L\\4000(\frac{1}{60})(\frac{\pi}{180})\)

Also, try \(\displaystyle \L\\2Rsin(\frac{\theta}{2})=2(4000)sin(\frac{\pi}{21600})\)

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Radians are absolute. There are 360 degrees in a circle because that's how we have chosen to divide up a circle(I think it came from the Babylonians). Actually, during WWII, they used something called a 'gradient'. You may still see it on calculators today. They thought American GI's were too stupid to understand 360 degrees in a circle so they used 400 degrees in a circle. The point is, radians stay constant because they are intrinsic to the circle. There are \(\displaystyle \frac{180}{\pi}\) degrees in a radian. No matter what. There are \(\displaystyle 2{\pi}\) radians in a circle, therefore, \(\displaystyle 2{\pi}\) radians in 360 degrees.

If there was something other than 360 degrees in a circle, then we would convert accordingly. But radians are constant. Mathematically, try getting used to them. Especially, in calculus.

 

[messa]

Oh thank you so much! I saw that formula pi/10800 but I had no idea where the number came from! You're such a big help. Thank you