Using Technology to teach Trigonometry By Dan Adamchick

Lots of students are taught Trigonometry in math class 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.

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!

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.

Here is a blueprint of a flange containing six bolt holes 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.

Notice that all hole dimensions will be off the center of the bolt circle, or X 0, Y 0.

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

2 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 1 The second hole is at location: X 0.866 Y 0.500

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. 3 The third hole is at location: X 0.866 Y -0.500

1 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. 4 The fourth hole is at location: X 0.000 Y –1.000

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.. 5 The fifth hole is at location: X –0.866 Y -0.500

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. 6 The sixth hole is at location: X –0.866 Y 0.500

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.

Good Job!

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.

Notice that all hole dimensions will be off the center of the bolt circle, or X 0, Y 0.

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

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 2 The second hole is at location: X 0.951 Y 0.309

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 3 The second hole is at location: X 0.588 Y – 0.809

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 4 The second hole is at location: X - 0.588 Y - 0.809

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 5 The second hole is at location: X - 0.951 Y 0.309

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.

Good Job!

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