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°