how to express radians to degrees, minutes, and seconds?
how to express fractions to degrees, minutes, and seconds?
through how many radians does the hour hand of a watch turn in 5 hours? 1/2 hour? 10 minutes? 3 days? between 8 am and 5:30 pm?
convert radians to degrees, as usual. Then multiply by 60 and 60 on the decimal parts:
23.461°
.461 * 60 = 27.66'
.66 * 60 = 40"
So, 23°27'40"
Do the reverse to go the other way. Divide by 60 and 3600 to convert ' and " to °.
Now use that knowledge along with the fact that 360° = 2pi radians.
To express radians to degrees, minutes, and seconds, you can use the following formulas:
1 radian = 180 degrees / π
1 degree = 60 minutes
1 minute = 60 seconds
Now, let's apply these formulas to answer your questions:
1. To express radians to degrees, minutes, and seconds:
- Multiply the given value in radians by (180/π) to get the value in degrees.
- Take the whole number part as degrees, then multiply the decimal part by 60 to get the minutes.
- Take the whole number part of the minutes as minutes, then multiply the decimal part by 60 to get the seconds.
2. To express fractions to degrees, minutes, and seconds:
- Convert the fraction to a decimal.
- Multiply the decimal equivalent by 360 to get the degrees.
- Take the whole number part as degrees, then multiply the decimal part by 60 to get the minutes.
- Take the whole number part of the minutes as minutes, then multiply the decimal part by 60 to get the seconds.
3. Let's calculate the number of radians the hour hand of a watch turns in:
a. 5 hours:
- The hour hand completes one revolution (360 degrees or 2π radians) in 12 hours.
- Divide the given time by 12, then multiply by 2π to get the answer in radians.
b. 1/2 hour:
- Divide the given fraction by 12, then multiply by 2π to get the answer in radians.
c. 10 minutes:
- Divide the given time (in minutes) by 720 (12 hours in minutes), then multiply by 2π to get the answer in radians.
d. 3 days:
- Multiply the given time by 24 (hours in a day), then multiply by 2π to get the answer in radians.
e. Between 8 am and 5:30 pm:
- Calculate the total time in minutes between the two given times.
- Divide this total by 720 (12 hours in minutes), then multiply by 2π to get the answer in radians.
With these formulas and explanations, you should be able to convert radians to degrees, minutes, and seconds and find the number of radians the hour hand of a watch turns for different durations.