Adding angles using the sexagesimal measure.

Both time and angles are measured using the sexagesimal measure. Lets look at the following problem:

Luis is a marathon runner who ran a marathon two days running as part of his training. He found the following: on the first day he ran the marathon in 2 h 48' 35''; and on the second day he ran it in 2 h 45' 30''. How long did Luis run for in total over the two days.

If we add up the hours, minutes and seconds separately we get the following:

  2h  48'  35"

+  2h  45'  30"  

4h  93'  65"

But 65 seconds is equal to one minute (60 seconds) and 5 seconds, so we could write the total as follows:

4h  94'  5"

Likewise, 94' is equal to one hour and 34 minutes. Therefore, the total is:

5h  34'  5"

We use the same process to add up the size of angles.

4. Add the following angles together in your exercise book:

a.     56º 20' 40"  +  37º 42' 15"

b.     125º 15' 30"  +  24º 50' 40"

c.     33º 33' 33"  +  17º 43' 34"

Use the following window to construct the angles and check your results.

d1, m1 and s1 correspond to the degrees, minutes and seconds of the first angle being added and d2, m2 and s2 correspond to those of the second angle being added.

Subtracting angles using the sexagesimal measure.

In the first race Luis' friend ran the marathon in exactly 3 hours. What is the difference between their two times?

We need to carry out the following operation:

3h   0'   0"

-  2h  48'  35"  

As with adding times, we need to subtract the hours, minutes and seconds separately. However, as we can't take 35 or 48 away from 0 we have to change one hour into 60 minutes and one minute into 60 seconds. Therefore, 3 hours is transformed into 2h 59' 60''.

2h  59'  60"

-  2h  48'  35"  

0h  11' 25"

5. Subtract the angles from the exercise above in your exercise book:

a.     56º 20' 40"  -  37º 42' 15"

b.     125º 15' 30"  -  24º 50' 40"

c.     33º 33' 33"  -  17º 43' 34"

Use this window to construct these angles and check your results.

d1, m1 and s1 correspond to the degrees, minutes and seconds of the first angle in the operation and d2, m2 and s2 correspond to those of the second angle being subtracted from the first one.

Multiplying an angle by a natural number.

In order to multiply an angle by a natural number we need to multiply each of the parts of the angle separately (degrees, minutes and seconds). If any of the products of second or minutes is greater than 60 we change it into one unit of the next bigger part.

18º  26'  35"

             *  3   

54º  78' 105"

But 105" = 1' 45", so

54º  79'  45"

But 79' = 1º 19', so

55º 19' 45"

6. Work out the following products:

a.     56º 20' 40" * 2

b.     37º 42' 15" * 4

c.     125º 15' 30" * 2

d.     24º 50' 40" * 3

e.     33º 33' 33" * 3

f.     17º 43' 34" * 2

Put the values for these angles and products into the following window and check your results for the exercise above.

In order to divide an angle by a natural number we need to divide the degrees by this number. Then we change the remainder into minutes, by multiplying by 60, and we add this number to the minutes we already have. We then divide the minutes. Then we change the remainder into seconds, by multiplying by 60, and we add this number to the seconds we already have. We then divide the seconds.

7. Divide the following angles by the natural numbers given:

a.     56º 20' 40" : 5

b.     37º 42' 15" : 4

c.     125º 15' 30" : 5

d.     25º 50' 40" : 6

e.     33º 33' 33" : 2

f.     17º 43' 34" * 2

Put the values for these angles and divisors into the following window and check your results for the exercise above.