Assume time t runs from zero to 2π and that the unit circle has been labled 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.

A. Starts at 12 o'clock and moves clockwise one time around.

B. Starts at 6 o'clock and moves clockwise one time around.

C. Starts at 3 o'clock and moves clockwise one time around.

D. Starts at 9 o'clock and moves counterclockwise one time around.

E. Starts at 3 o'clock and moves counterclockwise two times around.

F. Starts at 3 o'clock and moves counterclockwise to 9 o'clock.

1. x=/cos (2t); / y = / sin (2t)

2. x=/cos (t); / y = - / sin (t)

3. x=/sin (t); / y = / cos (t)

4. x=/cos{/frac{t}{2}}; / y = / sin{/frac{t}{2}}

5. x=-/cos (t); / y = - / sin (t)

Step-by-step explanation:

Given that, time runs from 0 to 2π.

Generally

x - direction, t makes an angle with x direction

x = Cos (t)

y = Sin (t)

y-direction, t makes an angle with y-direction

x = Sin (t)

y = Cos (t)

1. x = Cos (2t), y = Sin (2t)

Relating this to x = ACos (wt) and y=ASin (wt)

Where A is amplitude and w is angular frequency

Since w = 2 it shows that the moves counter clockwise two times round the clock

x = Cos (2t) implies that the clock is in the x direction i. e. at 3'0 clock since it is positive

Then, the match to this is

E. Starts at 3 o'clock and moves counterclockwise two times around.

2. x = Cos (t) and y=Sin (t)

So this has an angular frequency of

1, I. e. it moves clockwise round the clock ones

Now,

Since x = Cost, then, it is in positive x direction.

Then, the match to this is

C. Starts at 3 o'clock and moves clockwise one time around.

3. x = Sin (t) and y = Cos (t)

Also, the angular frequency is 1 and it moves clockwise one time round

Now, since y = Cos (t) it shows that it is positive y direction I. e. at 12'I clock.

Then, the match for this is

A. Starts at 12 o'clock and moves clockwise one time around.

4. x = Cos (½t) and y = Sin (½t)

Now, the angular frequency is ½. So, it doesn't move a full clockwise or counter clock wise revolution, it moves half revolution

Since x = Cos (½t), then, the position of the clock is in positive x - direction, i. e. at 3'o clock and it moves half revolution it will revolves to 9'o clock

So, the best match is

F. Starts at 3 o'clock and moves counterclockwise to 9 o'clock.

5. x = - Cos (t) and y = - Sin (t)

The angular frequency is 1 and it will moves counter clockwise one time round since it is negative.

Now, since x = - Cost, this shows that it is in x direction but negative x - direction i. e. at 9'o clock

So, the match for this is

D. Starts at 9 o'clock and moves counterclockwise one time around.