This part of the unit is really about equivalence:
12 = 1 ?
12 inches = 1 foot
5280 = 1
5280 feet = 1mile
C
=
d
Remember: π
y
1
C = 2π r
And because we are dealing with the unit circle here, we can say that for this special case, x
−1
1
C = 2π
−1
C
=
d
Remember: π
y
Recall the formula for arc length
30°
2π r
360°
1
C = 2π
r =1
30°
h
engt
arc l
Since 30° is 30° = 1
360° 12
of the way around,
then the length of
this arc is
−1
arc length =
x
1
1
1
C = 2π
12
12
1
π
C=
12
6
So for any radius r when
the central angle is 30°
the length of the arc is…
30° →
π
6
r
−1
arc length
π
= r
6
π
C
=
d
Remember: y
1
C = 2π
1
8
of the way around,
then the length of
this arc is
Since 45 is
r =1
45°
−1
arc length =
x
1
1
1
C = 2π
8
8
1
π
C=
8
4
So for any radius r
when the central
angle is 45
1
π
C= r
8
4
45° →
−1
arc length
π
= r
4
π
C
=
d
Remember: y
1
C = 2π
r =1
1
6
of the way around,
then the length of
this arc is
Since 60 is
60°
−1
arc length =
x
1
1
1
C = 2π
6
6
1
π
C=
6
3
So for any radius r
when the central
angle is 60
1
π
C= r
6
3
60° →
−1
arc length
π
= r
3
π
45° →
4
π
30° →
6
π
60° →
3
360° → 2π
We now can measure angles with a new unit of measurement: Radians
360° = 2π radians
Recall how to cancel units from science classes?
12 inches = 1 foot
5280 feet = 1mile
⎛ 1 ft ⎞
80 inches = 80 inches ⎜
⎟
12
inches
⎝
⎠
⎛ 5280 ft ⎞
2.3 miles = 2.3 miles ⎜
⎝ 1mile(s) ⎟⎠
= 6.75 ft
= 12,144 ft
We now have this ratio:
Converting degrees to radians
Multiply by
π rad
180°
2π radians π radians
=
360°
180°
Converting radians to degrees
Multiply by
180°
π rad
To sum this up, in PreCalc and beyond you’ll see how important radian measurements
are in math. The first direct application we’re seeing here is how they are the direct
way to translate angle measurements into arc lengths.
Convert to radians
Convert to degrees
π rad
π
60°
=
rad
180°
3
π 180
= 30°
6 π
π
π
= rad
180 2
2π 180
= 120°
3 π
90°
π
3π
135°
=
rad
180 4
5π 180
= 225°
4 π
π
10π
200°
=
rad
180
9
7π 180
= 210°
6 π
Notice that the radian problems on the right don’t have units. For reasons we won’t
discuss here, angle measurements without units are assumed to be radians.