DEGREEE MEASURE VS. RADIAN
MEASURE
Objectives:
To learn the difference between degree
measurement and radian
measurement.
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 2
R.
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 1 radian = 360/2 1 degree= 60 minutes 1 minute = 60
seconds
|
This
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,
then
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)
degrees
=
57 degrees
We also know that 1 complete revolution
is 2
radians, or approximately 6.283 radians.
Examples
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
120°=2/3
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 =
.45°.
And 1''
=
1'/60 = 1°/3600. Therefore,
25'' = 25°/3600 =
0.00694444°.
Hence, 54°27'25'' = 54.45694444.
EXERCISES
1. Convert 45 degrees
to radians.
2. Convert 5/9
radians to
degrees.
3 Convert 1/2 radian to
degrees.
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
degrees.
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
minutes?
10 How far does the tip of the
minute hand rotate in 35
minutes if the hand is 4 inches
long?
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?
_______________________________________________________________________
ANSWERS
1. ?/4
radians
2.
100°
3.
28.6479°
4. ?/3
meters
5. 5?/3
meters
6. 60?/3° = 62.85°
7. 45°27'0''
8. 45.7625°
9. 5?/3 radians
10. 7?/3 inches
11. 55o? Feet
12. 1/? inches