4-1 Review
1. Convert -4.2 radians to degrees
2. Order from smallest to largest:
1 revolution, 1 degree, 1 radian
4-1 Review
1. Convert -4.2 radians to degrees
-4.2 rad x ( 180°/Π rad ) =
-756°/ Π = -240°
2. Order from smallest to largest:
1 revolution, 1 degree, 1 radian
1 degree, 1 radian, 1 revolution
Quote of the Day
“A good hockey player plays
where the puck is. A great
hockey player plays where the
puck is going to be.”
-Wayne Gretzky
4-2 Lengths of Arcs and Areas of
Sectors
Circumference of a circle=2∏r = ∏d
Area of a circle= ∏r2
Example: Find the length of arc AB if
A
m<ACB = 75° and the radius
B is 6 cm.
C
C = (75/360) x ∏ x 12 = (5/2) ∏
Arc Length in Radians
Arc length in degrees = θ/360° ◦ 2∏r
(θ=angle measure in degrees, r=radius)
Converting this formula to radians…
Because 360° = 2∏, 360 and 2∏ cancel
in the formula. So,
Arc length in radians = θr
(θ = angle measure in radians, r=radius)
*Use the 1st formula for degrees and the
2nd one for radians.
Example
Find the length of an arc of a central
angle with measure ∏/3 radians in a
circle with radius 5 cm.
Arc length in radians = θr
= ∏/3 ◦ 5
= 5∏/3 cm
Area of a Sector
A = θ/360° ◦ ∏r2 (in degrees)
Convert to radians:
Knowing that ∏ radians = 180°, use
substitution
A = θ/360° ◦ 180r2 = 180 θr2 / 360
Reduce and get (180 goes into 360 twice):
Area of a sector in radians= θr2 / 2
Example of area of a sector
Find the area of a sector of a circle
with radius 10 cm and central angle
2∏/3 radians.
A = θr2 / 2
= (2∏/3) ◦ 100 / 2 = (2∏/3) ◦ 50
= 100 ∏/3 or 104.7 sq. cm
Exact
approximate
(Better answer)
How fast are we moving?
Earth travels around the sun in an elliptical
orbit, but one that is nearly circular with
radius 93,000,000 miles. Using the circular
approximation, about how far does the
Earth travel in a single day? Use 365.25
days for the length of one year.
1/365.25 revs ◦ 2∏ / 1 rev = 2∏ / 365.25 rad.
s = θr = 93,000,000 ◦ 2∏ / 365.25 =
=1,599,825.4 miles
How many miles per hour is that?
“Earth Rise”
Assignment
Do pages 242-243 1-4, 11-14, 18-20