6.3 Angles and Radian Measure
Section 6.3 includes three parts. Once all sets of notes and assignments are
completed, a graded assignment will be given. Here is the checklist with a pacing
guide to ensure you are staying on track to finish the section when needed.
Part 1 (Days 1-2)
o
o
Extending Angle Measure (example 1)
Arc Length and Radian Measure (example 2)
o
Converting Between Degrees and Radians (examples 3 and 4)

Assignment: pg. 441 # 1-33 odds
Part 2 (Day 3)
o
o
Arc Length and Angular Speed
 Arc Length (example 5)
 Central Angle Measure (example 6)
Assignment: pg. 441 # 47- 63 odds
Part 3 (Days 4-5)
o
o
Linear and Angular Speed (Example 7)
Assignment: pg. 442 # 69-83 odds
Days 6-7 will be a graded assignment of the section.
You will turn in your completed notes and assignments for a grade equal in size to
the graded assignment.
Extending Angle Measure (pg. 433- 434)
In an angle, the starting position of the ray is called the _______________________
and its final position after the rotation is called the__________________________.
An angle is said to be in standard position is its vertex is at the _____________ and its
initial side is on the _____________________________________.
Angles formed by different rotations that have the same initial and terminal sides are
called _____________________________.
Follow Example 1:
Find three angles coterminal with an angle of 600 in standard position.
Practice 1:
Find three angles coterminal with an angle of300 in standard position.
Arc Length (pg. 434- 435)
The length of an arc depends on the _____________________ of the circle and the
measure of the ____________________________ that it intercepts.
The length l of an arc is
l=
Example 2: Finding an Angle Given an Arc Length
An arc in a circle has an arc length l which is equal to the radius r. Find the measure of the
central angle that the arc intercepts.
Practice 2:
An arc in a circle has an arc length l which is equal to one- half of its radius r. Find the
measure of the central angle that the arc intercepts.
Radian Measure (pg. 435- 437)
Because it simplifies many formulas in calculus and physics, radians are used as a unit of
___________________________________ in mathematical and scientific applications.
Angle measurement in radians can be described in terms of the ____________________
which is the circle of radius 1 centered at the ___________________.
The Unit Circle
Converting Between Degrees and Radians (pg. 437- 438)
П radians = ____________0
Dividing both sides by П shows that 1 radian = ______________ ≈ _______________
Similarly, both sides of the original equation can be divided by 180.
To convert radians to degrees, multiply by
To convert degrees to radians, multiply by
Example 3: Converting from Radians to Degrees
Convert the following radian measurements to degrees.
Practice 3:
Convert the following radian measurements to degrees.
D.
𝜋
9
E.
3𝜋
5
F. 5𝜋
Example 4: Convert from Degrees to Radians
Convert the following degree measurements to radians.
a. 750
b. 2200
c. 4000
Practice 4:
Convert the following degree measurements to radians.
d. 1500
e. 3300
Assignment: pg. 441 # 1-33 odds
f. 5400
Arc Length and Angular Speed
The formula for arc length can also be written in terms of radians.
An arc with central angle measure θ radians has length
In other words, the arc length is the radius time the radian measure of the central
angle of the arc.
Example 5: Arc Length
The second hand on a clock is 6 inches long. How far does the tip of the second hand move
in 15 seconds?
solution: Every 60 seconds, the second hand makes a full revolution, or ________ radians.
During a 15 second interval it will make ___________ of a revolution, moving through an
angle of _______________________ radians, so the tip of the second hand travels
along an arc with a central angle measurement of ______. therefore, the distance that
the tip moves in 15 seconds is the arc length
l = rθ = ______________ = ____________ ≈__________ inches
Practice 5:
The second hand on a clock is 5 inches long. How far does the tip of the second hand move
in 45 seconds?
Example 6: Central Angle Measure
Find the central angle measure (in radians) of an arc of length 5 cm on a circle with a
radius of 3 cm.
solution:
Solve the arc length formula l = rθ for θ.
Practice 6:
Find the central angle measure (in radians) of an arc of length 11 cm on a circle with a
radius of 4 cm.
Assignment: pg. 441 # 47 – 63 odds
Linear and Angular Speed (pg. 439- 440)
Suppose that a wheel is rotating at a constant speed around its center. There are two
ways to measure the speed of a point on the outer edge of the wheel,
__________________________ speed and ______________________ speed.
linear speed = ___________ = _________
angular speed = __________ = _________
where θ is the ___________________________ of the angle through which the object
travels in time t. Notice that
linear speed = _____________________________
Example 7: Linear and Angular Speed
A merry-go-round makes 8 revolutions per minute.
a. What is the angular speed of the merry-go-round in radians per minute?
b. How fast is a horse 12 feet from the center traveling?
c. How fast is a horse 4 feet from the center traveling?
Practice 7:
A circular mobile above a baby’s crib makes 5 revolutions per minute.
a. What is the angular speed of the mobile in radians per minute?
b. How fast is a toy 9 inches from the center traveling in feet per minute?
c. How fast is a toy 6 inches from the center traveling in feet per minute?
Assignment: pg. 442 # 69- 83 odds