#$&* course Mth 164 11:30 am1/18/15
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Given Solution: The angular position changes by pi/6 radians every second. Starting at angular position 0, the angular positions at t = 1, 2, 3, 4, ..., 12 will be pi/6, 2 pi/6, 3 pi/6, 4 pi/6, 5 pi/6, 6 pi/6, 7 pi/6, 8 pi/6, 9 pi/6, 10 pi/6, 11 pi/6, and 12 pi/6. You might have reduced these fractions the lowest terms, which is good. In any case this will be done in the next problem. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q002. Reduce the fractions pi/6, 2 pi/6, 3 pi/6, 4 pi/6, 5 pi/6, 6 pi/6, 7 pi/6, 8 pi/6, 9 pi/6, 10 pi/6, 11 pi/6, and 12 pi/6 representing the angular positions in the last problem to lowest terms. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: pi / 6 2 pi / 6 = pi / 3 3 pi / 6 = Pi / 2 4 pi / 6 = 2 pi / 3 5 pi / 6 6 pi /6 = Pi 7 pi / 6 8 pi / 6 = 4 pi /3 9 pi / 6 = 3 pi /2 10 pi/ 6 = 5 pi /3 11 pi / 6 12 pi /6 = 2 pi confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe reduced fractions are pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q003. Sketch a circle centered at the origin of an x-y coordinate system, depicting the angular positions pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi. What are the angular positions of the following points: The point 2/3 of the way along the arc between (0,1) and (-1,0) The point 1/3 of the way along the arc from (0, 1) to (-1,0) The points 1/3 and 2/3 of the way along the arc from (-1,0) to (0,-1) The points 1/3 and 2/3 of the way along the arc from (0, -1) to (0,1)?? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The point 2/3 of the way along the arc between (0,1) and (-1,0) is: 5 pi /6. The point 1/3 of the way along the arc from (0, 1) to (-1,0) is: 2 pi /3 The points 1/3 and 2/3 of the way along the arc from (-1,0) to (0,-1) is: At 1/3 is 7 pi /6 and At 2/3 is 4 pi /3 The points 1/3 and 2/3 of the way along the arc from (0, -1) to (0,1) is: At 1/3 is 11 pi /6 and At 2/3 is pi /6 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe points lying 1/3 and 2/3 of the way along the arc between the points (0,1) and (-1,0) are at angular positions 2 pi/3 and 5 pi/6; the point 2/3 of the way between these points is at angular position 5 pi/6. The points lying 1/3 and 2/3 of the way along the arc between the points (-1,0) and (0,1) are at angular positions 7 pi/6 and 4 pi/3. The points lying 1/3 and 2/3 of the way along the arc between the points (0,-1) and (1,0) are at angular positions 5 pi/3 and 11 pi/6. Note that you should be able to quickly sketch and label this circle, which depicts the angles which are multiples of pi/6, whenever you need it. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): The final points are as follows: The points 1/3 and 2/3 of the way along the arc from (0, -1) to (0,1)?? Leading me to give the answers for these points instead of the points given in the solution which are: (0,-1) and (1,0) I believe this is a typo and I was on the right track to answering these questions. ------------------------------------------------ Self-critique Rating: 3
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Given Solution: `aThe angular position changes by pi/4 radians every second. Starting at angular position 0, the angular positions will be pi/4, 2 pi/4, 3 pi/4, 4 pi/4, 5 pi/4, 6 pi/4, 7 pi/4, and 8 pi/4. You might have reduced these fractions the lowest terms, which is good.In any case this will be done in the next problem. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q005. Reduce the fractions pi/4, 2 pi/4, 3 pi/4, 4 pi/4, 5 pi/4, 6 pi/4, 7 pi/4, and 8 pi/4 representing the angular positions in the last problem to lowest terms. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: pi / 4 2 pi / 4 = pi /2 3 pi/4 4 pi / 4 = pi 5 pi / 4 6 pi /4 = 3 pi / 2 7 pi / 4 8 pi /4 = 2 pi confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe reduced fractions are pi/4, pi/2, 3 pi/4, pi, 5 pi/4, 3 pi/2, 7 pi/4, and 2 pi. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q006. Sketch a unit circle (i.e., a circle of radius 1) centered at the origin of an x-y coordinate system, depicting the angular positions pi/4, pi/2, 3 pi/4, pi, 5 pi/4, 3 pi/2, 7 pi/4, and 2 pi. What are the angular positions of the following points: The point 1/2 of the way along the arc between (0,1) and (-1,0) The point 1/2 of the way along the arc from (0, -1) to (1,0) The point 1/2 of the way along the arc from (0,-1) to (0, -1)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The angular position 1/2 of the way along the arc between (0,1) and (-1,0) is: 3 pi / 4 The angular position 1/2 of the way along the arc from (0, -1) to (1,0) is: 7 pi / 4 The angular position 1/2 of the way along the arc from (0,-1) to (0, -1): Pi /2 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe point lying 1/2 of the way along the arc between the points (0,1) and (-1,0) (the topmost and leftmost points of the circle) is at angular position 3 pi/4. The point lying 1/2 of the way along the arc between the points (0,-1) and (1,0) is at angular position 7 pi/4. The point lying 1/2 of the way along the arc between the points (-1,0) and (0,-1) is at angular position 5 pi/4. These angles are shown in Figure 21. Note that the degree equivalents of the angles are also given. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I see the last coordinates do not line up for the question and the given solution but according to the given graph above my answers are correct based on the question. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q007. If the red ant starts at angular position pi/3 and moves at an angular velocity of pi/3 radians every second then what will be its angular position at the end of each of the first 6 seconds? Reduce your fractions to lowest terms. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: At second 1 it will be at pi /3 Second 2 2 pi /3 Second 3 3 pi / 3 = pi Second 4 4 pi /3 Second 5 5 pi / 3 Second 6 6 pi / 3 = 2 pi confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe angular position changes by pi/3 radians every second. Starting at angular position pi/3, the angular positions after successive seconds will be 2 pi/3, 3 pi/3, 4 pi/3, 5 pi/3, 6 pi/3 and 7 pi/3, which reduce to 2 pi/3, pi, 4 pi/3, 5 pi/3, 2 pi and 7 pi/3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I believe the answer 7 pi /3 is for the 7th second and is unnecessary since we only went for 6 seconds. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q008. Where is the angular position 7 pi/3 located? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I believe this is the same position as pi /3. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aIf you have not done so you should refer to your figure showing the positions which are multiples of pi/6. On your picture you will see that the sequence of angular positions 2 pi/3, pi, 4 pi/3, 5 pi/3, 2 pi, 7 pi/3 beginning in the first quadrant and moving through the second, third and fourth quadrants to the 2 pi position, then pi/3 beyond that to the 7 pi/3 position. The 7 pi/3 position is therefore identical to the pi/3 position. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q009. If the red ant starts at angular position pi/3 and moves at an angular velocity of pi/4 radians every second then what will be its angular position at the end of each of the first 8 seconds? Reduce your fractions to lowest terms. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I can look at my graph and see where these points would lie but I’m not positive on how to find this solution. confidence rating #$&*: 0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe angular position changes by pi/4 radians every second. Starting at angular position pi/3, the angular positions after successive seconds will be pi/3 + pi/4, pi/3 + 2 pi/4, pi/3 + 3 pi/4, pi/3 + 4 pi/4, pi/3 + 5 pi/4, pi/3 + 6 pi/4, pi/3 + 7 pi/4 and pi/3 + 8 pi/4. These fractions must be added before being reduced to lowest terms. In each case the fractions are added by changing each to the common denominator 12. This is illustrated for pi/3 + 3 pi/4: We first multiply pi/3 by 4/4 and 3 pi/4 by 3/3, obtaining the fractions 4 pi/12 and 9 pi/12. So the sum pi/3 + 3 pi/4 becomes 4 pi/12 + 9 pi/12, which is equal to 13 pi/12. The fractions add up as follows: pi/3 + pi/4 = 7 pi/12, pi/3 + 2 pi/4 = 5 pi/6, pi/3 + 3 pi/4 = 13 pi/12, pi/3 + 4 pi/4 = 4 pi/3, pi/3 + 5 pi/4 = 19 pi/12, pi/3 + 6 pi/4 = 11 pi/6, pi/3 + 7 pi/4 = 25 pi/12 and pi/3 + 8 pi/4 = 7 pi / 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I see where this solution is quite simple. I was probably overthinking the concept and missed a simple method. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q010. Starting at angular position pi/4 and moving at pi/3 radians / second what will be the resulting positions after each of the first 6 seconds? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Pi /4 + pi / 3 = 3 pi /12 + 4 pi /12 = 7 pi / 12 Pi / 4 + 2 pi/ 3 = 3 pi / 12 + 8 pi /12 = 11 pi / 12 Pi / 4 + 3 pi / 3 = 3 pi / 12 + 12 pi / 12 = 15 pi / 12 Pi / 4 + 4 pi /3 = 3 pi / 12 + 16 pi / 12 = 19 pi / 12 Pi / 4 + 5 pi/3 = 3 pi / 12 + 20 pi /12 = 23 pi / 12 Pi / 4 + 6 pi/3 = 3 pi / 12 + 24 pi / 12 = 27 pi / 12 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q011. Starting at angular position 0, the angular positions at t = 1, 2, 3, 4, ..., 12 will be pi/6, 2 pi/6, 3 pi/6, 4 pi/6, 5 pi/6, 6 pi/6, 7 pi/6, 8 pi/6, 9 pi/6, 10 pi/6, 11 pi/6, and 12 pi/6. So after 12 seconds we will have moved through an arc of 12 pi / 6 radians. Since 12 pi / 6 reduces to 2 pi, we will have moved through an arc of 2 pi radians, and we will be back at our starting point. If we continue to move around the circle for one more second we will have moved, in 13 seconds, through a total angle of 13 pi / 6 radians, and we will be at the same point on the circle as when we had moved through pi/6 radians. Through what total angle will we have moved by the end of each of the next 4 seconds, and at what previously visited point on the circle will we be located at the end of each? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: After 13 pi/6 which is the same position as pi /6 we have the following for the next 4 seconds: 14 pi / 6 = 7 pi / 3 which is at the same position as pi /3 15 pi / 6 = 5 pi / 2 which is at the same position as pi /2 16 pi / 6 = 8 pi / 3 which is at the same position as 2 pi / 3 17 pi / 6 which is at the same position as 5 pi / 6 The total angle for the position 17 pi / 6, if we’re talking about degrees can be easily found by doing the following: 17 pi/ 6 radians * 180 / pi = 3060 pi / 6 pi = 510 degrees for this total angle. You can see that the circle is divided by 30 degree angles since 360/ 12 is 30 this also makes it easy to see what the degrees will be by looking at the circle. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q012. This is an optional challenge question. If we start from angular position 0 and move through 7 pi / 4 radians every second, through what total angle will we have moved and where on the circle will we be at the end of each of the first 4 seconds? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The angle would be 1260 degrees circling around 3 ½ times stopping at the same point as 180 degrees or at pi. This position is at 7 pi. The end of the first 4 would be: 7 pi / 4 *1 second = 7 pi /4 7 pi / 4 * 2 seconds = 14 pi / 4 = 7 pi / 2 7 pi / 4 * 3 seconds = 21 pi / 4 7 pi / 4 * 4 seconds = 28 pi / 4 = 7 pi confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q010. Starting at angular position pi/4 and moving at pi/3 radians / second what will be the resulting positions after each of the first 6 seconds? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Pi /4 + pi / 3 = 3 pi /12 + 4 pi /12 = 7 pi / 12 Pi / 4 + 2 pi/ 3 = 3 pi / 12 + 8 pi /12 = 11 pi / 12 Pi / 4 + 3 pi / 3 = 3 pi / 12 + 12 pi / 12 = 15 pi / 12 Pi / 4 + 4 pi /3 = 3 pi / 12 + 16 pi / 12 = 19 pi / 12 Pi / 4 + 5 pi/3 = 3 pi / 12 + 20 pi /12 = 23 pi / 12 Pi / 4 + 6 pi/3 = 3 pi / 12 + 24 pi / 12 = 27 pi / 12 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q011. Starting at angular position 0, the angular positions at t = 1, 2, 3, 4, ..., 12 will be pi/6, 2 pi/6, 3 pi/6, 4 pi/6, 5 pi/6, 6 pi/6, 7 pi/6, 8 pi/6, 9 pi/6, 10 pi/6, 11 pi/6, and 12 pi/6. So after 12 seconds we will have moved through an arc of 12 pi / 6 radians. Since 12 pi / 6 reduces to 2 pi, we will have moved through an arc of 2 pi radians, and we will be back at our starting point. If we continue to move around the circle for one more second we will have moved, in 13 seconds, through a total angle of 13 pi / 6 radians, and we will be at the same point on the circle as when we had moved through pi/6 radians. Through what total angle will we have moved by the end of each of the next 4 seconds, and at what previously visited point on the circle will we be located at the end of each? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: After 13 pi/6 which is the same position as pi /6 we have the following for the next 4 seconds: 14 pi / 6 = 7 pi / 3 which is at the same position as pi /3 15 pi / 6 = 5 pi / 2 which is at the same position as pi /2 16 pi / 6 = 8 pi / 3 which is at the same position as 2 pi / 3 17 pi / 6 which is at the same position as 5 pi / 6 The total angle for the position 17 pi / 6, if we’re talking about degrees can be easily found by doing the following: 17 pi/ 6 radians * 180 / pi = 3060 pi / 6 pi = 510 degrees for this total angle. You can see that the circle is divided by 30 degree angles since 360/ 12 is 30 this also makes it easy to see what the degrees will be by looking at the circle. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q012. This is an optional challenge question. If we start from angular position 0 and move through 7 pi / 4 radians every second, through what total angle will we have moved and where on the circle will we be at the end of each of the first 4 seconds? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The angle would be 1260 degrees circling around 3 ½ times stopping at the same point as 180 degrees or at pi. This position is at 7 pi. The end of the first 4 would be: 7 pi / 4 *1 second = 7 pi /4 7 pi / 4 * 2 seconds = 14 pi / 4 = 7 pi / 2 7 pi / 4 * 3 seconds = 21 pi / 4 7 pi / 4 * 4 seconds = 28 pi / 4 = 7 pi confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating: #*&!