Using Technology

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Using Technology to teach Trigonometry
By Dan Adamchick
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Lots of students are taught Trigonometry in math
class. The students learn it by memorizing the
concepts. Hopefully they can grasp the concept
well enough and can use it to solve problems.
There is sine and cosine, Cartesian coordinates,
angles and right sides that everybody learns
about.
Education classes do not teach students how these
concepts apply in the real world often enough.
This presentation will show how Trigonometry is
applied to making things. It will give an
overview of how a machinist uses trig to make a
part with a milling machine.
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The Milling Machine
This is a milling machine. It is a machine tool
used to drill and bore holes, cut slots, and
create flat surfaces on material. If you are
familiar with a drill press, you can think of a
milling machine as a heavy duty drill press that
has a table that can move in three directions.
Work gets mounted to the table and the cutter
mounts in the spindle. You move the table with
the workpiece under the spinning cutter, and cut
away material. That is basically what a milling
machine (mill) is used for. In the hands of a
skilled machinist, a mill can do much more, such
as make angled surfaces, produce elliptical
surfaces, and even make gears. A mill can even
generate a sphere!
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DRO
Y
X
For this example of using Trigonometry on the
milling machine, we will concentrate on the table
movement, or the mills ability to locate holes.
Shown here are the X and Y directions of table
travel. The X and Y values for table position are
displayed on the digital readout or DRO. These
values relate directly to the Cartesian
coordinate system.
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Here is a blueprint of a flange containing six
bolt holes. This is typically all the
information that the engineer gives to the
machinist to make a part. Notice that only one
hole location is given, and all the others have
to be calculated or inferred. Given is only the
bolt hole circle radius. The machinist needs to
use Trigonometry to calculate these hole
locations.
Radius
The next series of slides will show how these
hole locations are calculated using Trigonometry.
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Notice that all hole dimensions will be off the
center of the bolt circle, or X 0, Y 0.
Center
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1
For the first hole, we see that the X value is
zero and the Y value is the radius. They are both
in a positive quadrant.
The first hole is at location X 0 Y 1.000
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Trig is as follows 360 / 6 60 X (SIN 60)
x 1.000 radius X 0.866 Y (COS 60) x 1.000
radius Y 0.500
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The second hole is at location X 0.866 Y 0.500
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1
For the third hole, we see that the X and Y
values are also the same as hole two. The X value
is in a positive quadrant and the Y value is in a
negative quadrant.
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The third hole is at location X 0.866 Y -0.500
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For the fourth hole, we see that the X value is
zero and the Y value is the radius, but it is a
negative quadrant.
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The fourth hole is at location X 0.000 Y
1.000
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1
For the fifth hole, we see that the X and Y
values are also the same as hole two. The X and Y
values are in the negative quadrant..
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The fifth hole is at location X 0.866 Y -0.500
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1
For the sixth hole, we see that the X and Y
values are also the same as hole two. The X value
is in a negative quadrant and the Y value is in a
positive quadrant.
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The sixth hole is at location X 0.866 Y 0.500
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Check marks indicate entered data.
A useful tool to help students visualize how the
Sine and Cosine functions relate to
Right-Triangle Trigonometry is Plane Triangle
Solver.
In this online application, you have to enter the
A) 60 degrees between holes, B) 90 degrees of the
right-triangle and b) 1 for the one inch radius.
After entering these three variables the program
calculates all the attributes of the triangle.
Notice that it solves for the X value of 0.866
and the Y value of 0.500. The key to using this
application is knowing how to enter the variables
to get the correct answer. The three-colored
triangles shown above represent how pieces of
known data were entered into the solver program.
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Good Job!
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Here is another blueprint of a flange, this time
containing five bolt holes. This is a typical
part drawing. Notice again that only one hole
location is given, and all the others have to be
calculated or inferred. Given is only the bolt
hole circle radius. The machinist needs to use
Trigonometry to calculate these hole locations.
Radius
The next series of slides will show how these
hole locations are calculated using Trigonometry.
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Notice that all hole dimensions will be off the
center of the bolt circle, or X 0, Y 0.
Center
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1
For the first hole, we see that the X value is
zero and the Y value is the radius. They are both
in a positive quadrant.
The first hole is at location X 0 Y 1.000
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Trig is as follows 360 / number of holes x
(hole number -1) 360 / 5 holes 72 72 x
(2nd hole - 1) 72 X (SIN 72) x 1.000
radius X 0.951 Y (COS 72) x 1.000 radius Y
0.309
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The second hole is at location X 0.951 Y
0.309
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Trig is as follows 360 / number of holes x
(hole number -1) 360 / 5 holes 72 72 x (3rd
hole - 1) 144 X (SIN 144) x 1.000 radius X
0.588 Y (COS 144) x 1.000 radius Y - 0.809
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The second hole is at location X 0.588 Y
0.809
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Trig is as follows 360 / number of holes x
(hole number -1) 360 / 5 holes 72 72 x (4th
hole - 1) 216 X (SIN 216) x 1.000 radius X
- 0.588 Y (COS 216) x 1.000 radius Y -
0.809
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The second hole is at location X - 0.588 Y -
0.809
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Trig is as follows 360 / number of holes x
(hole number -1) 360 / 5 holes 72 72 x (5th
hole - 1) 288 X (SIN 288) x 1.000 radius X
- 0.951 Y (COS 288) x 1.000 radius Y 0.309
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The second hole is at location X - 0.951 Y
0.309
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Another useful tool to help students visualize
how the Sine and Cosine functions relate to
Right-Triangle Trigonometry is Trigonometry
Realms.
In this online application, for hole 2 you
enter the D) 72 degrees between holes and f) 1
for the one inch radius. After entering these two
variables the program calculates all the
attributes of the triangle. Notice that it solves
for the X value of 0.588 and the Y value of
0.309. For Hole 3 is referencing off the 180
plane, so use 180 144 36. Enter 36 degrees
for the location of hole 3. The program will
then calculates X 0.588 and Y 0.809. Be sure to
obey the signs for each quadrant and plug the
calculated values into holes 4 and 5.
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Good Job!
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References
Abecedarical Systems - Free Mathematics Tutorials
and Software http//home.att.net/srschmitt/script
_plane_triangles.html Zona Land - Education in
Physics and Mathematics http//id.mind.net/zona/
theIndex/theIndex.html