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6.03 Degrees, minutes, and seconds


We've already looked at how to find angles and distances using trigonometric ratios, but have measured or rounded these angles to whole degrees. 

Often angles are involved in measuring values where accuracy is important, such as location or distance. So we need to be more accurate and use smaller units to measure angles.

When talking about parts of an angle that are less than a degree, one option would be to use decimal values. For example, we can measure an angle to be $34.56^\circ$34.56° which means it is between $34^\circ$34° and $35^\circ$35°. We can also use different units of measurement to describe the smaller parts of an angle, and these units are minutes and seconds.



Minutes and Seconds

There are $60$60 minutes in $1$1 degree. We write $a$a minutes as $a'$a.

There are $60$60 seconds in $1$1 minute. We write $b$b seconds as $b''$b.

The angle can then be written as $⬚^\circ\text{ }⬚'\text{ }⬚''$°  


Converting decimals to degrees and minutes

To convert a number in decimal form into minutes and seconds, we need first to find how much of $60$60 minutes the decimal part of the number represents.

For example, let's say we wanted to round $15.72^\circ$15.72° to the nearest minute.

We need to work out $15^\circ$15° + $0.72$0.72 of a degree, which is $15^\circ$15° plus $0.72$0.72 of $60$60 minutes.

$0.72\times60=43.2$0.72×60=43.2 minutes

Because we are rounding to the nearest minute, we then round the number of minutes to a whole number, which in this case is $43$43.

So, $15.72^\circ$15.72° rounded to the nearest minute is $15^\circ43'$15°43.


Converting decimals to degrees, minutes and seconds

But what if instead we wanted to round $15.72^\circ$15.72° to the nearest second? 

Writing $15.72^\circ$15.72° in terms of degrees and minutes (without rounding) gave us $15^\circ$15°$43.2'$43.2. To get the number of seconds we will need to look at the decimal part in the number of minutes and find out how many seconds out of $60$60 seconds it represents.

We need to work out $43'$43 + $0.2$0.2 of a minute, which is $43$43 plus $0.2$0.2 of $60$60 seconds.

$0.2\times60=12$0.2×60=12 seconds

So, $15.72^\circ$15.72° rounded to the nearest second is $15^\circ43'12"$15°4312".



Rounding values expressed in degrees and minutes is similar to rounding decimals. However, because there are $60$60 minutes in a degree, the half way point is $30$30:

  • If the number of minutes is less than $30$30, we round down to the nearest degree.
  • If the number of minutes is $30$30 or more, we round up to the nearest degree.

For example, when rounding to the nearest degree $148^\circ38'25"$148°3825" becomes $149^\circ$149° and $148^\circ27'25"$148°2725" becomes $148^\circ$148°.

The same principle is true for rounding seconds to minutes - since there are $60$60 seconds in a minute, the half way point is $30$30:

  • If the number of seconds is less than $30$30, we round down to the nearest minute.
  • If the number of seconds is $30$30 or more, we round up to the nearest minute.

For example, when rounding to the nearest minute $71^\circ13'45"$71°1345" becomes $71^\circ14'$71°14 and $71^\circ13'20"$71°1320" becomes $71^\circ13'$71°13.


Practice questions

Question 1

Write $38.38^\circ$38.38° in degrees, minutes and seconds.

Question 2

Find the acute angle $\theta$θ.

Round your answer to the nearest minute.


Question 3

Use the side lengths provided to find the angle $\theta$θ to the nearest minute.

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