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I want to know that converting from degree to radian and radian to degree is just understandable. But what is the purpose of expressing degree in term of minutes and seconds.. I know that there are 60 seconds in 1 minute and 60 minutes in one degree.. But still i am just wondering that where it is applicable to expressing degree along with minutes and seconds..

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Is your question a mathematical history question ("What's the history of base-60 being preferred to base-10?") or is it about present day usage? If the second, could you give examples of where you've encountered it? – Peter Taylor Mar 10 '14 at 11:08
In text book. we have to convert like 10°32'10'' into radian.. I know how to convert. but I want to know why we use minutes and seconds in degree??? – zonnie Mar 10 '14 at 11:14
@zonnie We subdivide one degree into 60 minutes, and one minute into 60 seconds. As stated in the comment above, we use here a base 60 arithmetic (just as we do nowadays to check the clock...). – DonAntonio Mar 10 '14 at 11:19
@zonnie.For your example :$32'$ is $32/60$ degrees; $10''$ is $10/60$ minutes, which then means $10/3600$ degrees. So $10°32'10''$ is $10+32/60+10/3600$ that is to say $3793/360$ degrees. Now, $180$ degrees is $\pi$ radians. Is this clear ? If not, just post. By the way, you could have asked the same question with $miles$, $yards$, $feet$ and $inches$; it is just the same problem. – Claude Leibovici Mar 10 '14 at 11:30
Thanx you all of u.. – zonnie Mar 10 '14 at 18:35

There's one situation in which minutes and seconds are remarkably useful: doing celestial navigation without a calculator. It turns out that the circumference of the earth at the equator is just about 360 * 60 miles; it's close enough that folks defined the "Nautical mile" to be about 7/6 of a land-mile, so that there are exactly 360 * 60 nautical miles around the earth at the equator. That means that each time you move one nautical mile east or west at the equator, your longitude changes by 1/60 of a degree (because there are 360 degrees of longitude around the earth). So at the equator, one minute of angle = one nautical mile in the east-west direction.

As it happens, the sun moves around the earth (from the point of view of the earthbound mariner, anyhow!) in 24 hours, so in one hour, it travels through 15 degrees of longitude (at least that's true at the spring equinox and fall equinox). That means that in one minute of time, it travels one quarter of a degree, or 15 minutes of angle. Note the conversion: one minute of time becomes 15 minutes of angle. That very simple relation between time (which happens to be divided into 60 minutes per hour and 60 seconds per minute) and angle (which we also divide into 60 minutes per degree and 60 seconds per minute) makes many of the computations we do for navigation much much simpler --- simpler enough that good approximations of them can be carried out in one's head, which is also helpful. (It also extends to the other units: one second of time becomes 15 seconds of angle; one hour of time becomes 15 degrees of angle.)

Of course, with the advent of calculators, doing everything in decimal makes a ton of sense.

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Thank you John.. I got my answer.. Once again thanx for giving such a nice description... – zonnie Mar 10 '14 at 18:32

About 5000 years ago, the Babylonians used a base-60 system for integers and fractions. The use of minutes and seconds to divide both hours and angles derives from this.

There is no inherent superiority to using minutes and seconds over using decimals. The use of minutes and seconds of arc in modern pedagogy is probably motivated primarily by traditionalism, and where it is considered necessary to defend it is probably defended by the argument that modern students need to be able to understand the concept when they read ancient texts.

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Thanks Peter... – zonnie Mar 10 '14 at 18:33

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