Expert: Paul Klarreich - 12/5/2007

Question
QUESTION:  What is the size, in degrees, of the angle subtended by an arc of 1 2/3 ft. in a circle whose radius is 45 in. ?

ANSWER: Questioner:   cristal
Category:  Advanced Math
Private:  No
 
Subject:  plane trigonometry
Question:  What is the size, in degrees, of the angle subtended by an arc of 1 2/3 ft. in a circle whose radius is 45 in. ?
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Hi, Cristal,

Two principles:

1. Radian measure defined:

         arc length
theta(in radians) = -----------
         radius


2. Converting radians to degrees: (d for degrees)

theta(in radians)     pi
----------------- = -----
theta(in degrees)    180d

OR:

theta(in degrees)    180d
----------------- = -----
theta(in radians)     pi




Apply 1:

1 2/3 feet = 20 inches, so
         20
theta =  --- = 4/9 radians.
         45

Apply 2:

theta(d)   180d
-------- = ----
4/9        pi
         80d
theta(d) = -----
         pi

Use your calculator; mine gives  25.464790894703253723021402139602, or
about  25.5 degrees.


---------- FOLLOW-UP ----------

QUESTION: can you express these in degrees, minutes and seconds:

4, 0.23, -1.4, 9.6, π/6, 3π/2

Answer
Questioner:   cristal
Category:  Advanced Math
Private:  No
 

---------- FOLLOW-UP ----------

QUESTION: can you express these in degrees, minutes and seconds:

4, 0.23, -1.4, 9.6, ð/6, 3ð/2
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Hi, again, Cristal,

To change each to degrees, apply Principle 2 to each.

SHORTCUT:

For the ones expressed in terms of pi, just replace the 'pi' symbol with 180d.


But for something like 4 (radians) do the proportion:

theta(in degrees)    180d
----------------- = -----
     4          pi

t = 720/pi =  229.18311805232928350719261925642

that's 229 degrees and change.  What change?  The fractional part can be converted to mins and seconds like this:

1. Subtract off the 229.  That leaves the fractional part.

.18311805232928350719261925642

2. Multiply by 60.

10.987083139757010431557155385241

that's 10 minutes, plus..

2. Subtract off the 10.

.987083139757010431557155385241

3. Multiply by 60. (again)

59.224988385420625893429323114453

that's 59 seconds.

You have  229d  10' 59"

That's the scheme for all of them.  Of course, pi/6, 3pi/2 come out exact:

pi/6 = 180/6 = 30 d

3pi/2 = 3(180)/2 = 270 d