Posted by Mark on Wednesday, April 30, 2008 at 10:39am.
Consider a starting time of 12:00.
Every increment of time creates a new triangle with sides of 7 and 11 and an included angle of 5.4m which can be solved using the Law of Cosines.
Where did the 5.4 m come from?
The minute hand of a clock moves 360/60 = 6º/minute or 6º/m.
The hour hand of a clock moves 30/60 = .50º/m.
For example, at 25 minutes after 12:00, the minute hand has rotated 6(25) = 150º while the hour hand has rotated .50(25) = 12.5º. Therefore, the two hands are now 150 - 12.5 = 137.5º apart.
In general, the angle µ between the two hands is 5.5m up to the point when they are 180º apart.
We therefore, now have a triangle with sides of 7 and 11 and an included angle of 137.5º.
Using the Law of Cosines, a^2 = b^2 + c^2 - 2abcosA or a^2 = 7^2 + 11^2 - 2(7)11(cos137.5) = 16.83cm.
What happens beyond the point where the two hands are 180º apart you ask?
When 5.5m is greater than 180º, you subtract the result from 360º to derive the angle between the two hands.
Thus, at 12:40, µ = 360 - 5.5(40) = 140º.
At 1:00, the two hands are 30º apart.
After 1:00, the distance between the tips of the hands continues to decrease until they are 4cm apart when the hour hand has rotated to 30 + .5m and the minute hand has rotated to 6m.
At this point 30 + .5m = 6m or m = 5.4545 minutes after 1:00, the point in time when the two hands are coincident.
You can now continues in the same manner as you did earlier from 1:5.4545 to the point where the two hands are again 180º apart when 6m - .5m = 180 or m = 32.7272 minutes after 1:00.
I'll let you continue from here.
A slight correction to the last expression result in my previous response.
After 1:00, the distance between the tips of the hands continues to decrease until they are 4cm apart when the hour hand has rotated to 30 + .5m and the minute hand has rotated to 6m.
At this point 30 + .5m = 6m or m = 5.4545 minutes after 1:00, the point in time when the two hands are coincident.
You can now continues in the same manner as you did earlier from 1:5.4545 to the point where the two hands are again 180º apart when 6m - (30 + .5m) = 180 or m = 38.1818 minutes after 1:00.
I'll let you continue from here.
find in degrees and radians the angle between the hour hand and the minute hand of a clock at half past three
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