Transcript 4 1 Radians Degrees 01

Objectives:
1. To define angles and
their parts and
positions
2. To use radian and
degree measures for
angles
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Assignment:
P. 290: 1-5 (Some)
P. 290: 7-12 (Some)
P. 290: 13-15
P. 291: 25-30 (Some)
P. 291: 31-34 (Some)
P. 291: 35, 36
P. 293: 117-120
Homework Supplement
For the Greeks,
trigonometry dealt with
the side and angle
measures of triangles,
practical math for
building stuff and locating
the position of a
particular constellation
for accurately navigating
their boats and dubiously
predicting their future.
(The Greeks doing trig in fancy dress.)
(An expensive piece of equipment that
your iPhone could probably duplicate for
99 cents.)
For those later influenced by
Greek mathematics,
trigonometry deals with
periodic functions, whose
graphs are usually pretty
wavy, and whose
applications far exceed the
polygonal constraints of
triangles: periodic motion,
sound, and other
interesting stuff.
(Trig is instrumental in solving crimes.)
For those later influenced by
Greek mathematics,
trigonometry deals with
periodic functions, whose
graphs are usually pretty
wavy, and whose
applications far exceed the
polygonal constraints of
triangles: periodic motion,
sound, and other
interesting stuff.
You will be able to
define angles and
their parts and
positions
Recall from geometry that
an angle is made from
two rays joined at a
common endpoint
called a vertex.
We usually use the Greek
letters α (alpha), β
(beta), and θ (theta) to
label an angle. Or
maybe just a capital
letter.
In trigonometry, an
angle is formed by
rotating one ray (the
initial side) to the
other ray (the terminal
side).
It is often convenient and
fun to place angles in a
rectangular coordinate
system.
An angle is in standard
position if its vertex is on
the origin and its initial
side is on the positive xaxis.
An angle is positive if
the terminal side is
rotated counterclockwise.
An angle is negative if
the terminal side is
rotated clockwise.
You will be able to use
radians and degree
measures for angles
The measure of an angle is the amount of
rotation from the initial side to the terminal
side as measured in degrees or radians.
0  mA 

2
mA 


2
2
 mA  
mA  
The curious among you may
be wondering why there
are 360 degrees in a
circle, while others may
not even care. The
answer is actually pretty
simple: It’s because there
are 360 days in the year.
At least that’s what the
Babylonians thought, and
they are the ones who
came up with the crazy
idea called a degree.
Each year, of course, is
made up of 12
“months.” Further,
each of those
“months” is made up
of 30 “days.” 12
times 30 equals 360
degrees, I mean days.
1. On a clock, how
many degrees does
the hour hand rotate
each hour?
2. How many degrees
does the minute
hand rotate each
minute?
Which angle has a larger measure, ABC
orDBE?
In geometry, we
always measured
angles by the
smallest amount of
rotation from the
initial side to the
terminal side, from
0° to 180°.
That’s because we
were mostly
dealing with
triangles and stuff.
But we can just as
easily have angles
with measures
greater than 180°,
just ask Danny
Way.
That’s because we
were mostly
dealing with
triangles and stuff.
But we can just as
easily have angles
with measures
greater than 180°,
just ask Danny
Way.
We can totally
have angles
greater than
180°.
So now, let’s build
ourselves a circle with
various degree
measures from 0 to
360. Yes, before you
ask, you will have to
have these memorized.
1. Start by drawing a
circle.
2. Now break the circle
into fourths by
adding an x- and yaxis. Label the points
of intersection 0, 90,
180, and 360.
3. Next, add your clock
hours. We already
have 3, 12, 9, and 6.
Just put two equally
spaced marks
between each
quarter. Label these
marks with multiples
of 30.
4. Finally, add a mark in
the middle between
30 and 60. This is 45.
Now add your
multiples of 45.
So that concludes the
degrees you should
have memorized.
For the adventurous,
you might try
memorizing
multiples of 15, too.
Sketch each of the following angle
measurements. To do so, start by drawing a
circle with the x- and y-axes, and then put
your angle into standard position.
1.
2.
3.
4.
60
135
210
270
5.
6.
7.
8.
−15
−150
−275
−315
The measure of an arc
is the measure of the
central angle it
intercepts. It is
measured in degrees.
An arc length is a
portion of the
circumference of a
circle. It is measured
in linear units and
can be found using
the measure of the
arc.
• Arc measure = mC
– Amount of rotation
• Arc length:
mC
s
 2 r
360
– Actual length
Here’s an interesting
question: If you
were to take the
radius of a circle
and wrap it
around the circle’s
circumference,
how far would it
reach?
1. Use a ruler to
draw a radius
from the center
of the circle to
the “3.” This is
like the initial
side of an angle
in standard
position.
2. Now cut a thin
strip of paper
from the bottom
edge of your
paper and mark
the length of the
radius of your
circle on the left
side of the strip.
3. Carefully wrap
this length along
the
circumference of
the circle and
mark it with
your pencil.
4. Use your ruler to
connect this
mark to the
center of the
circle with
another radius.
This is the
terminal side of
an angle we’ll
call θ.
The arc that intercepts
θ has length 1
radian, so we say
the measure of
θ = 1 radian.
Approximately how
many degrees is 1
radian?
Now let’s see how
many radians it
takes to span the
circle.
5. Use your ruler to
draw a diameter
from “9” to “3.”
This is like the xaxis.
6. Now use a
compass to
measure the
radian arc length.
Copy this length
around your circle
multiple times
until you have
gone (nearly) all
the way around.
You should notice
that it takes a
little bit more
than 3 radians to
span a
semicircle. In
fact, it takes
exactly π (≈3.14)
radians.
Also notice that it
takes a bit more
than 6 radians to
span the full
circle, which is
exactly 2π
(≈6.28) radians.
This should make
sense since the
circumference of
a circle is 2πr,
where r is the
radius of the
circle.
One radian is the measure of a central angle θ
that intercepts an arc s equal in length to the
radius r of the circle.
This time, we’re going
to build a circle with
various radian
measures from 0 to
2π.
1. Start by drawing a
circle.
2. Now break your
circle into fourths by
adding an x- and yaxis. Since a
semicircle is π, the
quarter circle is π/2.
Now count by
halves.
3. Next, add your clock
hours. We already
have 3, 12, 9, and 6.
Just put two equally
spaced marks
between each
quarter. Since there
are 6 of these marks
on the semicircle,
count by sixths.
4. Finally, add a mark
in the middle
between π/6 and
π/3. Since there
would be four of
these along the
semicircle, these
must be fourths.
So that concludes the
radians you should
have memorized.
For the adventurous,
you might try
memorizing the
twelths, too.
Sketch each of the following angle
measurements. To do so, start by drawing a
circle with the x- and y-axes, and then put
your angle into standard position.
1.
2.
3.
4.
π/3
3π/4
−π/2
−11π/6
5.
6.
7.
8.
−π/12
−2π
3 radians
−4.5 radians
Objectives:
1. To define angles and
their parts and
positions
2. To use radian and
degree measures for
angles
•
•
•
•
•
•
•
•
Assignment:
P. 290: 1-5 (Some)
P. 290: 7-12 (Some)
P. 290: 13-15
P. 291: 25-30 (Some)
P. 291: 31-34 (Some)
P. 291: 35, 36
P. 293: 117-120
Homework Supplement