4.2 degrees and radians...4.2 degrees and radians geometry trigonometry ray ray vertex vertex...
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4.2 Degrees and Radians
Geometry Trigonometry
ray
ray
vertex vertex initial side
terminal side
Angles in Standard PositionThe 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:
Convert 62.381° to DMS form
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 15.712° to DMS form
What part of an hour is .381°?
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!!!
1 RADIAN = arc length (s)radius (r)
Converting between Degrees & Radians
Full circle rotation = 360°
Full circumference of a circle = 2πr
sr
So, a full circle (360 degrees) expressed in radians = arc lengthradius
2πr r = 2π radians360° =
360° = 2π radians
180° = π radians
or
To convert degrees to radians, multiply by
To convert radians to degrees, multiply by
π180
π180
a) Convert 120° to radians
b) Convert 45° to radians
c) Convert 225° to radians
d) Convert to degrees5π6
e) Convert to degrees7π4
θ θ = sr
a) Convert 210° to radians
b) Convert -135° to radians
Warmup
Coterminal Anglesangles that "land" or "end" in the same position
120°Identify two coterminal angles with 120°
Lesson 4.2 Continued
Coterminal Angles = θ ± 360°
or θ ± 2π radians
Identify 3 coterminal angles:
a) 65°
b) -π3
Arc Length - the circumference of part of a circle
Arc Lengths = r·θ
(If θ is in radians)
Find the arc length "s"
45°10 cms 18 cm
s
Area of a Sector
Find the area of the blue sector
s = 2πr·
(If θ is in degrees)
θ360°
Arc Length
π6
4.2 HWp. 238 #1-5 odds, 10-20 evens, 27-28, 45-49 odds, 67
Find the area of the blue sector(s)
9 in2π3
12 cm80°
36°
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