Assume time t runs from zero to 2 pi/and that the unit circle has been labeled a
ID: 2878877 • Letter: A
Question
Assume time t runs from zero to 2 pi/and that the unit circle has been labeled as a clock. Match each of the pairs of parametric equations with the best description of the curve from the following list. Enter the appropriate letter (A, B, C, D, E, F) in each blank. Starts at 12 o'clock and moves clockwise one time around. Starts at 6 o'clock and moves clockwise one time around. Starts at 3 o'clock and moves clockwise one time around. Starts at 9 o'clock and moves counterclockwise one time around. Starts at 3 o'clock and moves counterclockwise two times around. Starts at 3 o'clock and moves counterclockwise to 9 o'clock. x = cos t/2;| y = sin t/2| x = sin(t);| y = cos(t)| x = cos(t);| y = -sin(t)| x = -sin(t);| y = -cos(t) x = cos(2t);| y = sin(2t);|Explanation / Answer
First, orient yourself. 0 degrees correspond to 3 o'clock, 90 deg to 6 o'clock, 180 deg to 9 o'clock, 270 deg to 12 o'clock.
Then, work out what happens to x and y right at the beginning, at t = 0.
The key is that the clock has two parts. x is the part of the clock in the horizontal direction (if y=0, then x=1 is 3 o'clock, x=-1 is 9 o'clock), and y is the vertical part (if x=0, y=1 is 12 o'clock, y=-1 is 6 o'clock)- you have to look at both two parts to get the final motion.
What do we have? For case (2), x is at zero, but y is at 1. Imagine the x-y axis - the horizontal direction is x, the vertical direction is y. So we have 0 in the horizontal direction, but 1 in the vertical direction. Therefore the result is something that points straight up, at 12 o'clock. We look for the option that starts at 12 o'clock - it is A.
2 -> A
Now look down to (4). Same thing, but now y points straight down, to 6 o'clock. B starts at 6 o'clock.
4 -> B
Now look at (1). We have the expression (t/2) inside the sine and cosine. That means at t = 2 pi, for example, we have sin(Pi) and cos(Pi). In other words, the clock is going twice as slow as t. Therefore it is the option where the clock only moves half a circle, from 3 to 9 o'clock - it is too slow to move a full circle.
1 -> F
Same thing for (5). Now the clock moves twice as fast as t. It moves two times round - option E.
5 -> E
Only one to go! In (3), the clock starts at 9 o'clock. It then moves counterclockwise - why? Because at t=Pi, for example, we get -sin(Pi) and -cos(Pi). Now, cos(Pi)=-cos(Pi), so that's okay, but the -sin(Pi) moves the clock in the opposite direction from t.
3 -> D