#$&* course Mth 164 Question: `q001. Note that this assignment has 9 activities.
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Given Solution: The angular position changes by pi/6 radians every second. Starting at angular position 0, the angular positions at t = 1, 2, 3, 4, ..., 12 will be pi/6, 2 pi/6, 3 pi/6, 4 pi/6, 5 pi/6, 6 pi/6, 7 pi/6, 8 pi/6, 9 pi/6, 10 pi/6, 11 pi/6, and 12 pi/6. You might have reduced these fractions the lowest terms, which is good. In any case this will be done in the next problem. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did not reduce the fractions. ------------------------------------------------ Self-critique rating #$&* 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, pi/3, pi/2, 2pi/3, 5pi/6, pi, 7pi/6, 4pi/3, 3pi/2, 5pi/3, 11pi/6, 2pi confidence rating #$&*Ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe reduced fractions are pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* 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: I sketched out the unit circle on paper. The point lying 2/3 of the way along the arc between (0,1) and (-1,0) is at position 5pi/6. The point lying 1/3 of the way along the arc between (0,1) and (-1,0) is at position 2pi/3. The point lying 1/3 of the way along the arc between (-1,0) and (0,-1) is at position 7pi/6. The point lying 2/3 of the way along the arc between (-1,0) and (0,-1) is at position 4pi/3. The point lying 1/3 of the way along the arc between (0,-1) and (0,1) is at position 5pi/3. The point lying 2/3 of the way along the arc between (0,-1) and (0,1) is at position 11pi/. confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe points lying 1/3 and 2/3 of the way along the arc between the points (0,1) and (-1,0) are at angular positions 2 pi/3 and 5 pi/6; the point 2/3 of the way between these points is at angular position 5 pi/6. The points lying 1/3 and 2/3 of the way along the arc between the points (-1,0) and (0,1) are at angular positions 7 pi/6 and 4 pi/3. The points lying 1/3 and 2/3 of the way along the arc between the points (0,-1) and (1,0) are at angular positions 5 pi/3 and 11 pi/6. Note that you should be able to quickly sketch and label this circle, which depicts the angles which are multiples of pi/6, whenever you need it. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok ********************************************* Question: `q004. If the red ant 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The ant starts at 0 and goes up pi/4 every second so: 0, pi/4, 2pi/4, 3pi/4, 4pi/4, 5pi/4, 6pi/4, 7pi/4, 8pi/4 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe angular position changes by pi/4 radians every second. Starting at angular position 0, the angular positions will be pi/4, 2 pi/4, 3 pi/4, 4 pi/4, 5 pi/4, 6 pi/4, 7 pi/4, and 8 pi/4. You might have reduced these fractions the lowest terms, which is good.In any case this will be done in the next problem. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Again I did not reduce the fractions. ------------------------------------------------ Self-critique rating #$&*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, pi/2, 3pi/4, pi, 5pi/4, 3pi/2, 7pi/4, 2pi confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe reduced fractions are pi/4, pi/2, 3 pi/4, pi, 5 pi/4, 3 pi/2, 7 pi/4, and 2 pi. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* Ok ********************************************* Question: `q006. Sketch a circle 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 point lying 1/2 of the way along the arc between (0,1) and (-1,0) is at position 3pi/4. The point lying 1/2 of the way along the arc from (0,-1) to (1,0) is at position 7pi/4. The point lying 1/2 of the way along the arc from (0, -1) to (0,-1) is at position 5pi/4. confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe point lying 1/2 of the way along the arc between the points (0,1) and (-1,0) (the topmost and leftmost points of the circle) is at angular position 3 pi/4. The point lying 1/2 of the way along the arc between the points (0,-1) and (1,0) is at angular position 7 pi/4. The point lying 1/2 of the way along the arc between the points (-1,0) and (0,-1) is at angular position 5 pi/4. These angles are shown in Figure 21. Note that the degree equivalents of the angles are also given.  &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok ********************************************* 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: The ant starts at 0 and goes up pi/3 every second so: 0, pi/3, 2pi/3, 3pi/3, 4pi/3, 5pi/3, and 6pi/3 Lowest terms: 0, pi/3, 2pi/3, pi, 4pi/3, 5pi/3, and 2pi confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe angular position changes by pi/3 radians every second. Starting at angular position pi/3, the angular positions after successive seconds will be 2 pi/3, 3 pi/3, 4 pi/3, 5 pi/3, 6 pi/3 and 7 pi/3, which reduce to 2 pi/3, pi, 4 pi/3, 5 pi/3, 2 pi and 7 pi/3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok ********************************************* Question: `q008. Where is the angular position 7 pi/3 located? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The 7pi/3 position will be in the same exacted position of pi/3. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aIf you have not done so you should refer to your figure showing the positions which are multiples of pi/6. On your picture you will see that the sequence of angular positions 2 pi/3, pi, 4 pi/3, 5 pi/3, 2 pi, 7 pi/3 beginning in the first quadrant and moving through the second, third and fourth quadrants to the 2 pi position, then pi/3 beyond that to the 7 pi/3 position. The 7 pi/3 position is therefore identical to the pi/3 position. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*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: The ant starts out at pi/3 and goes up pi/4 every second. We will need to add the pi/3 to each pi/4 it goes up every second so: pi/4+pi/3, 2pi/4+pi/3, 3pi/4+pi/3, 4pi/4+pi/3, 5pi/4+pi/3, 6pi/4+pi/3, 7pi/4+pi/3, 8pi/4+pi/3. to add these together there must be a least common denominator and then factor to the lowest terms so: 7 pi/12, 5 pi/6, 13 pi/12, 4 pi/3, 19 pi/12, 11 pi/6, 25 pi/12, 14 pi / 3. confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe angular position changes by pi/4 radians every second. Starting at angular position pi/3, the angular positions after successive seconds will be pi/3 + pi/4, pi/3 + 2 pi/4, pi/3 + 3 pi/4, pi/3 + 4 pi/4, pi/3 + 5 pi/4, pi/3 + 6 pi/4, pi/3 + 7 pi/4 and pi/3 + 8 pi/4. These fractions must be added before being reduced to lowest terms. In each case the fractions are added by changing each to the common denominator 12. This is illustrated for pi/3 + 3 pi/4: We first multiply pi/3 by 4/4 and 3 pi/4 by 3/3, obtaining the fractions 4 pi/12 and 9 pi/12. So the sum pi/3 + 3 pi/4 becomes 4 pi/12 + 9 pi/12, which is equal to 13 pi/12. The fractions add up as follows: pi/3 + pi/4 = 7 pi/12, pi/3 + 2 pi/4 = 5 pi/6, pi/3 + 3 pi/4 = 13 pi/12, pi/3 + 4 pi/4 = 4 pi/3, pi/3 + 5 pi/4 = 19 pi/12, pi/3 + 6 pi/4 = 11 pi/6, pi/3 + 7 pi/4 = 25 pi/12 and pi/3 + 8 pi/4 = 14 pi / 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok"