Download 4.2 Degrees and Radians

4.2 Degrees and Radians
Geometry
ray
vertex
Trigonometry
terminal side
ray
vertex
initial side
Angles in Standard Position
The measure of an angle equals the
amount of rotation required to move from
the initial side to the terminal side
Positive angle = counter clockwise
Negative angle = clockwise
1 degree = 1/360 rotation
90 degrees = 1/4 rotation
180 degrees = 1/2 rotation
270 degrees = 3/4 rotation
360 degrees = 1 full rotation
Another way of expressing degrees is in "DMS" Form:
= Degrees-Minutes-Seconds
Here's how:
Leave the whole number (that's the degrees)
Multiply the decimal part by 60 (that's the minutes)
Multiply the remaining decimal part by 60 again (that's the seconds)
Convert 62.381° to DMS form
What part of an hour is .381°?
Convert 15.712° to DMS form
Now go backwards. Convert 43° 11' 40" to degrees.
DMS format is usually only useful in navigation and surveying.
The mathematical standard for expressing degrees is in RADIANS!!!
arc length (s)
radius (r)
1 RADIAN =
θ
s
s
θ= r
r
Converting between Degrees & Radians
Full circle rotation = 360°
Full circumference of a circle = 2πr
So, a full circle (360 degrees) expressed in radians =
360° =
2πr
r
= 2π
arc length
radius
radians
360° = 2π radians
or
180° = π radians
To convert degrees to radians, multiply by
To convert radians to degrees, multiply by
a) Convert 120° to radians
b) Convert 45° to radians
c) Convert 225° to radians
d) Convert
e) Convert
5π
6
7π
4
to degrees
to degrees
π
180
180
π
Warmup
a) Convert 210° to radians
b) Convert -135° to radians
Lesson 4.2 Continued
Coterminal Angles
angles that "land" or "end" in the same position
120°
Identify two coterminal angles with 120°
Coterminal Angles = θ ± 360°
or
θ ± 2π radians
Identify 3 coterminal angles:
a) 65°
b)
-π
3
Arc Length -
the circumference of part of a
circle
Arc Length
Arc Length
s = r·θ
θ
s = 2πr· 360°
(If θ is in degrees)
(If θ is in radians)
Find the arc length "s"
s
10
cm
45°
s
π
6
Area of a Sector
Find the area of the blue sector
18 cm
Find the area of the blue sector(s)
36°
9 in
2π
3
12
c
80° m
4.2 HW
p. 238 #1-5 odds, 10-20 evens,
27-28, 45-49 odds, 67