DEGREEE MEASURE VS. RADIAN
To learn the difference between degree
measurement and radian
Recall 1: Think about how you could measure an angle. There are at least two ways: one is by degrees or the "wideness" of the angle, the other is by the length of the circle that it scribes. Look at the two illustrations below.
Recall 2 You probably recall that the circumference (total distance around) a circle of radius R is 2R. Of course, this means that the circumference of a circle of radius 1 is simply 2.
Recall 3 The total number of degrees around a circle is 360 degrees.
Recall 4 If we wish to measure angles by the length of the arc of circle that they cut off, then we need to have a name for the unit that describes such an angle. We call it radians. That is, angles are measured by the length of the arc of a circle of radius 1; that measurement unit is called radians.
In the illustration above, the angle measured is b radians.
Recall 5: From what we recall above,
we know that
360 degrees = 2 radians
1 degree = 2/360 radians
1 radian = 360/2 degrees
1 degree= 60 minutes
1 minute = 60
gives us a way of converting angle measurement
from degrees to radians
and radians to degrees.
For example, 1 degree equals 2/360 radians and 1 radian = 360/(2) degrees.
If you want to know how many radians are in 60 degrees just multiply both sides of the equation 1 degree = 2/360 radians by 60 to get 60 degrees = /3 radians.
Recall 6: The length of an arc of a circle of radius R, cut by an angle of b radians is bR. Now do you see why radian measure is so useful?
Recall 7: At times it is convenient to divide each degree into 60 parts, called minutes, and each minute into 60 parts, called seconds .
Minutes and seconds are often used in navigation. Minutes are written with a prime symbol, as in 34', which means 34 minutes or 34/60 of a degree. Seconds are written with a double prime symbol, as in 17'', which means 17 seconds, or 17/60 of a minute.
Introduction 1 (Area)
How can we find the area of a segment of a circle of radius R that is cut by an angle of b radians. See the illustration below. Can we find the area A of the shaded segment?
We know that the full area of a circle of radius R is R^2. We can set up a proportion relationship. The fraction of shaded area A is to the total area of the circle as b is to 2. This means that A/(R^2) = b/(2). This simplifies to A = (R^2)(b/2).
Keep in Mind
1. Place an angle at the center of a circle of radius 1 unit. If the angle cuts off an arc of length 2 units, then the angle has a measure of 2 radians.
In general, if the angle cuts off an arc of length b units,
the angle has a measure of b radians.
2. If you measure angles in the clockwise direction, then the angle has a measure that is equal to minus the length of the arc that it cuts. See the illustration below.
3. The size of a radian is given by the conversion between degrees and radians. We know that
1 radian = 360/(2)
= 57 degrees
We also know that 1 complete revolution is 2 radians, or approximately 6.283 radians.
Example 1. Degree measure is denoted by the symbol x°, which means x degrees. Some angle conversions are very handy to know. They are
0° = 0 radians
30°= /6 radians
45°= /4 radians
60°= /3 radians
90°= /2 radians
135°= 3/4 radians
150°= 5/6 radians.
Example 2. Convert 5/4 radians to degrees.
Solution: 1 radian = 180/ degrees.
Multiply both sides of this last equation by 5/4 to get
5/4 radians = (5/4)(180/) degrees. Now simplify the right hand side to get
5/4 radians = 225 degrees
Example 3. Convert degrees to radians.
Solution: 1 degree = /180 radians. Multiply both sides by to get degrees = ^2/180 radians, or approximately 1/20 of a radian.
Example 4. A pendulum of length 4 meters swings through an angle of 10 degrees on either side of a vertical line. How long is the path of the swing of the bob at the end of the cable?
Solution: The cable moves through a total angle of 20 degrees as is swings from side to side. The length of the path of the bob is simply the arc of the circle of radius 4 cut by an angle of 20 degrees. First convert the degree measurement to radian measurement.
1 degree = /180 radians, so 20 degrees = /9 radians.
The arc length is the radius times the angle as measured in radians. So arc length = 4(/9) meters.
Example 5. Write 54°27'25'' as a decimal.
Solution: 54°27'25'' = 54° + 27' + 25''.
But 1' = 1°/60. Therefore,
27' = 27°/60 =
And 1'' = 1'/60 = 1°/3600. Therefore, 25'' = 25°/3600 = 0.00694444°.
Hence, 54°27'25'' = 54.45694444.
1. Convert 45 degrees
2. Convert 5/9
3 Convert 1/2 radian to
4 Find the length of an arc
cut from a circle of radius
1 meter by an angle of 60 degrees placed at
the center of the circle.
5 Find the
length of an arc cut from a circle of radius
5 meters by an angle of 60
degrees placed at the center of the circle.
6 A 5-centimeter arc is cut by an angle at the center
of a circle
of radius 3. Find the measurement of the angle in
7 Write 45.45° using
degrees, minutes and seconds.
8 Write 45°45'45'' as a decimal.
9 Through how many
radians does the minute hand of a clock
rotate in 50
10 How far does the tip of the
minute hand rotate in 35
minutes if the hand is 4 inches
11 A 10-inch diameter wheel turns
at 100 revolutions per
minute. How many feet does a point on the
perimeter of the wheel
travel in 6 minutes?
12 A bug clings to the wheel in exercise #11 and travels 1000 feet in 5 minutes. How far is the bug from the center of the wheel?
6. 60?/3° = 62.85°
9. 5?/3 radians
10. 7?/3 inches
11. 55o? Feet
12. 1/? inches
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