Question

Can someone please tell me the mistake i made in this problem:
x=3cost and y=3sint
i used this equation x^2+y^2=1
(3cost)^2+(3sint)^2=9
the final answer is cos^2t +sin^2t=9(but i got 1)

Answer 1

the equation for a circle is x^2 + y^2 = r^2. You have to find r yourself by plugging x and y into the equation.

\[(3\cos t)^2+(3\sin t)^2=9\cos^2t+9\sin^2t\]
\[9(\sin^2t+\cos^2t)=r^2\]
9 = r^2
r = 3

Plug r = 3 in the general equation for the circle. My preference would be to leave it as x^2 + y^2 = 9. However, you must substitute x = cost and y = sint because the radius is already factored in.

Answer 2

ty i understand it now. but now i am stuck on a similar problem you helped me previously with. i have to find when t<0 for the follwowing equations 10+t y=2t

Answer 3

ty i understand it now. but now i am stuck on a similar problem you helped me previously with. i have to find when t<0 for the follwowing equations 10+t y=2t

Answer 4

* x=10+t and y=2y

Answer 5

Solve for t in both equations. Then substitute them both in t<0. Know that you have to do x and y separately. Then, solve for x and y in their respective inequalities.

Answer 6

do i have to plug in zero fpr t when i solve for their equations

Answer 7

Right away, plug in 0 for t in both equations to find what point we're talking about. You should get x = 10 and y = 2 for the point (10, 2). To derive the inequality, solve for t without plugging 0s in.

Answer 8

yah i got the answer but since its less than it is left (10,0) ?

Answer 9

hey i dont know why my response shows up 10 times

Answer 10

Yeah, to the left of (10, 0) because x<10. You're right about the point, it is at (10, 0), not (10, 2)...my fault.

Answer 11

no worries thanks again for ur help

Answer 12

i have another question if you do not mind helping. i have to find the the parametrization of the unit circle or part of it for the following problem x=cos(t^2) and y=sin(t^2)

Answer 13

This one is parametrized. Is there anything specific being asked for?

Answer 14

i have another question if you do not mind helping. i have to find the the parametrization of the unit circle or part of it for the following problem x=cos(t^2) and y=sin(t^2)

Answer 15

it also asks how the circle is traced ou (clockwise and counterclockwise

Answer 16

It is traced along the unit circle because if you put x and y into the equation of a circle x^2 + y^2 = r^2, then you'll get r = 1, or x^2 + y^2 = 1. The manner in which it is traced is what makes it different than just an ordinary x = cost, y = sint. It traces in a counterclockwise direction just like a normal parametrization of the unit circle. However, it traces a lot faster. Do you know what I mean by faster?

Answer 17

the graph actually moves clockwise wen t is less than zero and increases when t is greater than zero

Answer 18

Oh sorry, I didn't consider negative t values. When t is negative the x = cost and y = sint parametrization goes clockwise. However, I would think that x = cos(t^2) and y = sin(t^2) would still go counterclockwise when t is negative because the negative would just cancel itself out...

Answer 19

i mean counterclockwise when it increases

Answer 20

how did you figure out the direction

Answer 21

The ones related to the unit circle are obvious for me because I know that's the way the unit circle moves (1, 0) when t = 0, (0, 1) when t = pi/2, (-1, 0) when t = pi, etc.

However, the best way to do it is to set up a table with t, x, and y. You choose t values, and then get corresponding x's and y's. It's crucial that you pick t values in numerical order (0, 1, 2, etc.) because the order to plot the points indicates the order they come in the path. t=0 comes before t=1 and t=1 comes before t =2.

Answer 22

but for this problem the t is squared so what i do? bc if i plug in pi/2 into cos it become cos(pi(^2))/4

Answer 23

Yeah, that's a yucky value to work with. What you do is pick a t value such that t^2 = n where n is a "nice number" like pi or pi/2. Just take the square root of t to get t = sqrt(n). So, think of a nice number, then use the square root of it.

Answer 24

my teacher used another method using the derivative
if the function is increasing then the derivative>0 so itcs counterclockwise if its dcreasing it clockwise

Answer 25

Haha, I didn't even think about the derivative. Yes, that works for many unit circle problems.

Answer 26

what type of problems specifically - when dealing with squared functions

Answer 27

You have to make sure you know which direction is the "increasing" direction - the orientation of the curve - when t is increasing. For example, the orientation of the curve x = cos(-t^2) and y = sin(-t^2), the derivative is positive when moving in a clockwise direction.

Answer 28

u mean counterclockwise

Answer 29

Hmm, now I'm just confused. I guess this is why I prefer to determine orientation numerically. Yes, the example I gave wasn't a good example... you'll have to ask your teacher about derivatives...haha.

Answer 30

what happens to the values on the unit circle when you move clockwise

Answer 31

The t values? T values would be negative, though the graphing calculator freaks out when you try to use them. I generally just use positive values of t when working with parameters, unless an interval of t with negative limits is given to me.

Answer 32

Or the t values would just go down, I guess would be a better way to put it.

Answer 33

in genral on the unit circle if u move clockwise what happens to the values do they become negative

Answer 34

hey just wanted to say thank you for everything

Answer 35

Well if you chose negative values of t with increasing absolute value (-1, -2, -3), you will be plotting points moving the clockwise direction. This isn't the correct way to think about orientation, though because we plot values of t of increasing value (-3, -2, -1, etc.).

I may be beginning to steer you in a bad direction. In general, I'd say not to think too much into negative values of t besides plotting them. Plotting points does not do anything for changing orientation. The orientation of the unit circle defined as x = cost and y = sint always moves in the counter clockwise direction.

If you're confused on what the orientation is, though you can graph the set of parametric equations on a graphing calculator. It traces it out with respect to the orientation.