January 07, 2016
January 07, 2016
(4.1 Day 2) Angles and Their Measure
Objective: 1. To learn the problem with angular measure.
2. To learn what a radian is.
3.
To convert between degrees and radians.
4. To learn the arc length formula for radian measure.
Why: Angles are the domain elements of trigonometric functions.
January 07, 2016
Obj. 1: To learn the problem with angular measure.
Read the last half of Pg.320 (The Problem of Angular Measure)
January 07, 2016
January 07, 2016
ACTIVITY:
Obj. 2: To learn what a radian is.
Constructing a 1 - Radian Angle
Materials: direction sheet, piece of paper, compass,
scissors, pipe cleaner, straightedge, pencil
Part I:
1. Use a compass and draw a large circle on a piece of paper.
2. Identify the circle (O) and draw a radius horizontally
from O toward the right, intersecting the circle at point A.
3. Cut a piece of pipe cleaner the same size as the radius.
(Be very precise with the length of the pipe cleaner being
exactly the same as the radius length)
4. Place one end of the pipe cleaner at A and bend it around
the circle counterclockwise, marking the point B on the circle
where the other end of the pipe cleaner ends.
5. Draw the radius from O to B. Shade
The measure of
AOB.
AOB is one radian.
6. In your own words, describe the measure of one radian:
_______________________________________________
_______________________________________________
Part II:
1. Label a point, C, on the circle where the m AOC is 2 radians.
2. Label a point, D, on the circle where the m AOD is 3 radians.
3. Label a point, E, on the circle where the m AOE is 4 radians.
5. Label a point, F, on the circle where the m AOF is 5 radians.
6. Label a point, G, on the circle where the m AOG is 6 radians.
Questions:
1. What is the formula for the circumference of the circle, in
terms of its radius, r?
1. _________
2. How many radians must there be in a complete circle?
2. _________
January 07, 2016
Obj. 2: To learn what a radian is.
Defn: Radian
A central angle of a circle has measure 1 radian if it
intercepts an arc with the same length as the radius
r
r
θ=1 radian
r
θ = 1 radian
if arc length = radius
January 07, 2016
Obj. 2: To learn what a radian is.
y
2 radians
1 radian
3 radians
x
6 radians
4 radians
5 radians
January 07, 2016
Obj. 2: To learn what a radian is.
Question:
About how many degrees is 1 radian?
2 radians?
3 radians?
4 radians?
5 radians?
6 radians?
Obj. 3: To convert between degrees
and radians.
January 07, 2016
Obj. 3: To convert between degrees and radians.
Make a proportion for 1 revolution of a circle:
radians
degrees
=
Converting:
Radians to Degrees: multiply by
Degrees to Radians: multiply by
180
π
π
180
January 07, 2016
Obj. 3: To convert between degrees and radians.
Convert the degrees to radians.
1. 45o
2. 270o
3. 330o
Convert the radians to degrees.
π
4.
6
5.
2π
3
6. 4.5 radians
January 07, 2016
Obj. *To learn the arc length formula for radian measure.
s
r
θ
Arc Length (Degree Measure):
Example:
s = πrθ
180
Arc Length (Radian Measure):
s = rθ
1. Find the radius of a circle that has an arc length of 2.5cm with central angle
π
.
that measures
3
January 07, 2016
Obj. 3: To convert between degrees and radians.
January 07, 2016
Objective: 1. To learn the problem with angular measure.
2. To learn what a radian is.
3. To convert between degrees and radians.
HW:
(if time) * To learn the arc length formula for radian measure.
(Reg): (4.1)Pg.325: 9-23odd, 27, 29, 30