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 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 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

= 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

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

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.

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?

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2. 100°

3. 28.6479°

4. ?/3 meters

5. 5?/3 meters

6. 60?/3° = 62.85°

7. 45°27'0''

8. 45.7625°