Question

the parametric curve has the equations
x=sin2t , y=cos2t

why does it move in the clockwise direction on the circle? and why does it go around twice?

Answer 1

Why? :d
Like you want a philosophical reasoning? :D
Because ummm ... if it wasn't so.... math... wouldn't... exist...

That's my way of saying, "I don't know"

Answer 2

But in all seriousness,
you can just plug in different t values to see the direction and stuff, ya?

Answer 3

\[\large\rm t=0:\qquad (x,y)=(0,1)\]\[\large\rm t=\frac{\pi}{4}:\qquad (x,y)=(1,0)\]\[\large\rm t=\frac{\pi}{2}:\qquad (x,y)=(0,-1)\]

Answer 4

Recall that in your normal unit circle,
cosine generally corresponds to the x-direction, while
sine corresponds to the y-direction

Since it's backwards in our parameter,
that must... be ... doing something.. ya

Answer 5

oh right, thanks

Answer 6

Yeah also that 2 in there tells you the frequency. So like if you go \(t=2 \pi\) normally that's one revolution for \(\sin 0 \to \sin 2 \pi\) right? But once you have that 2 in there, you only have to go for half as much time to get a full rotation, see:

\[\sin(2t)\]\( t = \pi\)

Answer 7

if you assume that
x = cos(2t) , y = sin(2t) moves counterclockwise from (1,0)
then x = sin(2t) , y = cos(2t) flips the roles of x and y, and geometrically you have the reflection about the line y = x.
The reflection about the line y = x changes the direction of clockwise/counterclockwise.

Also note that since x = cos(2t) , y = sin(2t) starts at (1,0) when t = 0 going ccw,
x = sin(2t) , y = cos(2t) starts at (0,1) and moves cw .

Answer 8

but why does it go around twice?

Answer 9

because domain of t is from 0 to 2pi
you can see it as \(x^2 = sin^2(2t) \\y^2 = cos^2 (2t) \\x^2 +y^2 =1\)

That is unit circle and domain \(0\leq t \leq 2\pi\)
Put it into the table like what @zepdrix did, you can see that why it goes around twice

Answer 10

The answers above are good. I will have a go at it.

We are looking at x = sin(2t), and y = cos(2t)
Let's analyze this expression. I am going to assume that t stands for time.

To simplify matters, lets rewrite the parametric equations as
x = sin(θ), y = cos (θ)
But the angle θ is actually equal to 2*t, which is twice the value of time t.
As time t goes from 0 to 2π each angle is double the value of time.
This causes the graph to complete 2 revolutions in the time it takes for
'x = cos(t), y = sin(t)' to complete one revolution.
For example
When t = π, the angle is 2*π. So you have completed one revolution when t = π.
When t = 2π, the angle is 4π , so you completed a second revolution when t= 2π