If your solution to stated problem does not match the given solution, you should self-critique per instructions at

 

   http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

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Your solution, attempt at solution.  If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

 

 

001.  Radian measure and the unit circle.

 

Goals for this Assignment include but are not limited to the following:

 

1.  Know the definition of the radian.

2.  Relate coordinate positions on the unit circle to angular displacement and to arc displacement.

 

Click once more on Next Question/Answer for a note on Previous Assignments.

 

 

 

Previous Assignments: 

 

Be sure you have completed all Preliminary Assignments as instructed on under the Assts link on the homepage.  These assignments include the q_a_orientation, and the three sets Initial Problems, Describing Graphs and Typewriter Notation from the q_a_init_pbs program.  Links and explanations are included on the Assignments Page.

 

Question:  `q001.  Note that there are 10 activities in this assignment.

 

Figure 37 (located under the Figures link on the Assignments page under Assignment 0) depicts a circle of radius 1 centered at the origin of a x y coordinate system. Imagine the we have 2 ants, one red and one black. Both start out moving at the same speed from the point for the positive x-axis beats the circle. The red ant crawls along the arc of the circle in the counterclockwise direction, and black ant crawls along the x-axis toward the origin. The ants proceed until the black ant reaches the origin. Both ants will have crawled the same distance, the black ant along a straight line and the red ant along an arc of the circle.

 

At that instant the red ant will have traveled a distance equal to 1 radius of the circle, and we say that the red ant has completed 1 radian of arc. Which of the indicated points on the circle will correspond to a 1 radian arc? Note that we have indicated points a, b, c, d.

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aWe see visually that the point a lies at an arc distance less than the radius of the circle. We also see that the point c lies at an arc distance that is clearly greater than the radius of the circle. The only possible candidate for a 1 radian angle, which must lie at an arc distance equal to one radius, is therefore point b.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q002. If the first ant moves at a constant speed, moving through 1 radian every second, then approximately how long, to the nearest second, do you think it will take for the ant to move along the arc to the point where the circle meets the negative x-axis?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aVisual examination, perhaps accompanied by a quick sketch, shows that it takes approximately 3 arcs each of one radian to get from the positive x-axis to the negative x-axis when moving along the arc of the circle.

 

In figure 37 the points b, c and d lie at approximately 1, 2 and 3 radians.  Remember that each radian corresponds to an arc distance equal to the radius of the circle.

 

At 1 radian / second it will take about 3 seconds to move the approximately 3 radians to the negative x axis.

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q003.  If the ant traveled at 1/2 radian per second, then after 1 second would its angular position be indicated by point a, point b, point c or point d in Figure 37?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aAfter 1 second the angular position would be 1/2 radian, which would correspond to point a.

 

Note that after 2 seconds the angular position would be 1 radian, corresponding to point b, and after three seconds the angular position would be 3 * 1/2 radian = 3/2 radian and the ant would be halfway between position b (which is at 1 radian) and position c (which is at 2 radians).

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q004.  How far will the ant travel in the process of completing 1 trip around the circle, starting and ending at the initial point where the circle meets the positive x-axis.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aThe circumference of the circle is 2 pi r, where r is the radius of the circle. This is the distance traveled by the ant.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q005.  As we just saw the distance around the circle is its circumference 2 pi r, where r is the radius. Through how many radians would the ant travel from the initial point, where the circle meets the positive x-axis, if the motion was in the counterclockwise direction and ended at the original point after having completed one trip around the circle.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aAn arc displacement of r corresponds to an arc distance of 1 radian on the circle.  Arc distances of 2, 3, 4, ... time the radius would correspond to 2, 3, 4, ... radians of arc.  That is, arc distance of r, 2r, 3r, 4r, ... correspond to 1, 2, 3, 4, ... radians of arc.

 

We understand by these examples that if we divide the arc distance by the radius, we will get the number of radians of angular distance.

 

The arc distance around the circle is 2 pi r, which therefore corresponds to 2 pi r / r = 2 pi radians.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q006.  The unit circle is a circle of radius 1 centered at the origin.  What are the coordinates of the points where the unit circle meets the positive x-axis, the positive y axis, the negative x-axis and the negative y axis?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:  

`aThe unit circle has radius 1 and is centered at the origin, so the circle meets the positive x-axis 1 unit from the origin at (x, y) = (1,0). Similarly the circle meets the positive y-axis at the 'top' of the circle, 1 unit from the origin at (x, y) = (0,1); the circle meets the negative x-axis at (-1, 0); and the circle meets the negative y-axis at (0,-1).

 

Figure 84 shows these points on the unit circle.  Note that in this figure the small dots are located at increments of .1 unit in the x and y directions.

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q007.  Without looking at Figure 84, sketch a picture of the unit circle, complete with labeled points where the circle meets the x and y axes.  Indicate the arc from the standard initial point, where the circle meets the positive x-axis, to the point where the circle meets the positive y axis.  Describe your sketch.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aYour sketch should show the x and y axes and a circle of radius 1, with the points (1,0), (0, 1), (-1, 0) and (0, -1) where the circle meets the coordinate axes labeled.   The arc will run along the first quadrant of the circle from (1,0) to (0,1).   Your figure should match figure 84.

 

You should be able to quickly draw this picture any time you need it.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q008.  How many radians of angular displacement correspond to the arc displacement from the standard initial point, where the circle meets the x-axis, to the point where the circle meets the positive y axis?  

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aThe trip around the entire circle, which corresponds to an angular displacement of 2 pi radians, corresponds to a trip from the initial point to the point where the circle meets the positive y-axis (i.e., the point (0,1)), then from this point to the point where the circle meets the negative x-axis (i.e., the point (-1,0)), then from this point to the point where the circle meets the negative y-axis (i.e., the point (0,-1)), then from this point back to the point where the circle meets the positive x-axis (i.e., the point (1,0)).

 

Because of the symmetry of the circle, the arc corresponding to each of these displacements is the same. The arc from (1,0) to (0,1) is 1/4 of the 2 pi radian angular displacement around the entire circle, so its angular displacement is 2 pi/4 = pi/2 radians.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q009.  We have just seen that the angular position of the (1,0) point is 0 and the angular position of the (0,1) point is pi/2.  What are the angular positions of the (-1,0) and (0,-1) points?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aThese points are reached after successive angular displacements of pi/2.  The (-1,0) point is reached from the pi/2 position by an additional angular displacement of pi/2, which puts it at angular position pi. 

 

The (0,-1) point is reached after another angular displacement of pi/2, which puts it at pi + pi/2 = 2 pi/2 + pi/2 = 3 pi/2.

 

Note that still another angular displacement of pi/2 puts us back at the initial point, whose angular position is 0.  This shows that the initial point has angular position 0, or angular position 3 pi/2 + pi/2 = 4 pi/2 = 2 pi, consistent with what we already know.

 

You should label your picture with these angular positions pi/2, pi, 3 pi/2 and 2 pi specified at the appropriate points.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q010.  What is the angular displacement from the standard initial point of the point halfway along the arc of the circle from (1,0) to (0,1)?  Note that you should begin with a sketch of the circle and of the arc specified here.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`a(1,0) is the point at which the circle meets the positive x-axis and (0,1) is the point at which the circle meets the positive y-axis. The trip along the arc of the circle from (1,0) to (0,1) will move along the first-quadrant arc from angular position 0 to angular position pi/2. Halfway along this arc, the angular position will be 1/2 * pi/2 = pi/4.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q011.  What will be the angular positions of the arc points halfway between the (0,1) and (-1,0) points of the circle?

 

What will be the angular positions of the arc points halfway between the (-1,0) and (0,-1) points of the circle?

 

What will be the angular positions of the arc points halfway between the (0,-1) and (1,0) points of the circle?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aHalfway between the (0,1) point, which corresponds to the the the position pi/2, and the (-1,0) point, which corresponds to angular position pi, will be the point lying at angular position pi/2 + pi/4 = 2 pi / 4 + pi / 4 = (2 pi + pi)/4 = 3 pi / 4.

 

Halfway between the (-1,0) point, which corresponds to the the position pi,and the (0,-1) point, which corresponds to angular position 3 pi / 2, will be the point lying at angular position pi + pi/4 = 4 pi / 4 + pi / 4 = (4 pi + pi)/4 = 5 pi / 4.

 

Halfway between the (0,-1) point, which corresponds to the the position 3 pi/2, and the (-1,0) point, which corresponds to angular position 2 pi, will be the point lying at angular position 3 pi/2 + pi/4 =62 pi / 4 + pi / 4 = (6 pi + pi)/4 = 7 pi / 4.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `q012.  What is the angular position of the point lying 1/3 of the way along the arc of the circle between the points (1,0) and (0,1)?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

`aThe arc from (1,0) to (0,1) corresponds to an angular displacement of pi/2. One-third of the arc corresponds to an angular displacement of 1/3 * pi/2 = pi/6. The angular position of the specified point is therefore pi/6.

 

 

 

Question:  `q013.  How long would it take to move around a complete circle at each of the following angular speeds:

 

 

Your solution: 

 

Confidence Assessment:

 

Question:  `q014.  What is the angular position in radians of the point which lies 3/4 of the way along the arc of the circle which starts at (0, 1) and runs counterclockwise to the point (0, -1).

 

Your solution: 

 

Confidence Assessment:

Question:  `q015.  What is the angular position in radians of the point which lies 2/3 of the way along the arc of the circle which starts at (0, 1) and runs counterclockwise to the point (-1, 0)?

 

Your solution: 

 

Confidence Assessment:

 

Self-critique Rating: