Question

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)

Answer 1

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.