Degrees on Unit Circle

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kcordell11

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"Consider a regular 12 hour clock, and measure time in hours starting from noon. Let m denote the angle that the minute hand makes with the top position (12). Similarly, let h be the angle that the hour hand makes with the vertical position. In this case, suppose we measure the angle clockwise, so that the angles involved are positive.
At 2:30pm we have"
h = 75 degrees
At that time the minute hand has completed two and half rotations, and so the angle
m = 900 degrees. (Enter a number between 720 and 1080.)
In general, at time t
h = ? degrees, and
m = ? degrees.
There is a time t between 2 and 3 pm at which the minute hand is on top of the hour hand. At that time the hour hand forms an angle x of ____?____ degrees with the vertical.
[I've seen the same question posted by another student but the response did not help, how do I go about solving this?]
 
how did you get 75 degrees for the hour hand at t = 2:30 = 2.5 hours after noon?

how did you get 900 degrees for the minute hand at t = 2:30 = 2.5 hours after noon?

how would you calculate for t = 3:15?
 
What are two good functions to use for cyclical phenomena?

Which one might be easier to work with if t = 0 gives an angle of 0 degrees?
 
skeeter said:
how did you get 75 degrees for the hour hand at t = 2:30 = 2.5 hours after noon?

how did you get 900 degrees for the minute hand at t = 2:30 = 2.5 hours after noon?

how would you calculate for t = 3:15?
Click to expand...
If a unit circle has a complete rotation of 360 degrees and there are 12 hours that equally divide into a regular 12 hour clock, I created this equation: 360/12, which equals 30 degrees. From there, I numbered each hour 30 degrees. Since the time is 2:30pm, the hour hand must be between 2 and three since the minute hand is halfway around the 12 hour (unit circle) clock. Counting from 2 to 3 is 1 integer, so half of that 1 (since the hand is between 2 and 3) is .5, now my new equation will be 2.5x30, which equals 75. That's how I got my first answer for the hour hand.

I got 900 degrees for the minute hand since one full rotation is 360 degrees and the time starts at 12 but now it's 2:30 (2 and a half (.5)). The minute hand completes 2 full rotations, 360x2=720, and the 30 minutes is half the rotation of 360, 360/2=180, so my new equation is 720+180=900 degrees for the minute hand.

How I would calculate if t=3:15 is saying h for the hour hand would point 1/4 the way between 3 and 4. 1/4 of an hour is = .25, which also = 15 minutes, hence 3:15. So the hour is 3+.25=3.25. Now since each hour represents 30 degrees, I will multiply 3.25 by 30 and get 97.5 degrees the angle the hour hand makes.
For the minute hand, ironically, on the other hand, It will point directly on 3 since the 3 on a clock symbolizes 15 minutes, if I count by fives. My new equation will be (3 hours is 3 rotations around a circle that is 360 degrees) 360x3=1080 degrees. I will also add the 15 minutes to it (where the minute hand is pointing, which is 3 makes a 90 degree angle from 12.) My new equation will be 1080+90=1170.
 
JeffM said:
What are two good functions to use for cyclical phenomena?

Which one might be easier to work with if t = 0 gives an angle of 0 degrees?
Click to expand...
I have no idea how to set up the next problem or what you said. Sorry.
 
kcordell11 said:
If a unit circle has a complete rotation of 360 degrees and there are 12 hours that equally divide into a regular 12 hour clock, I created this equation: 360/12, which equals 30 degrees. From there, I numbered each hour 30 degrees. Since the time is 2:30pm, the hour hand must be between 2 and three since the minute hand is halfway around the 12 hour (unit circle) clock. Counting from 2 to 3 is 1 integer, so half of that 1 (since the hand is between 2 and 3) is .5, now my new equation will be 2.5x30, which equals 75. That's how I got my first answer for the hour hand.

I got 900 degrees for the minute hand since one full rotation is 360 degrees and the time starts at 12 but now it's 2:30 (2 and a half (.5)). The minute hand completes 2 full rotations, 360x2=720, and the 30 minutes is half the rotation of 360, 360/2=180, so my new equation is 720+180=900 degrees for the minute hand.

How I would calculate if t=3:15 is saying h for the hour hand would point 1/4 the way between 3 and 4. 1/4 of an hour is = .25, which also = 15 minutes, hence 3:15. So the hour is 3+.25=3.25. Now since each hour represents 30 degrees, I will multiply 3.25 by 30 and get 97.5 degrees the angle the hour hand makes.
For the minute hand, ironically, on the other hand, It will point directly on 3 since the 3 on a clock symbolizes 15 minutes, if I count by fives. My new equation will be (3 hours is 3 rotations around a circle that is 360 degrees) 360x3=1080 degrees. I will also add the 15 minutes to it (where the minute hand is pointing, which is 3 makes a 90 degree angle from 12.) My new equation will be 1080+90=1170.
Click to expand...

so, can you come up with a general equation for $m$ and $h$ for any time $t$?
 
kcordell11 said:
I have no idea how to set up the next problem or what you said. Sorry.
Click to expand...
The angles are proportional to time.
So:
h is to t as 30 degrees is to 1 hour, therefore, h = ?
m is to t as ?, therefore, m = ?
 
lev888 said:
The angles are proportional to time.
So:
h is to t as 30 degrees is to 1 hour, therefore, h = ?
m is to t as ?, therefore, m = ?
Click to expand...
h = 30t
m = 360t

Thank you so much for your help. Now that I look at it, it was this simple but I was thinking too much, too hard about it. :)
 
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