Parametric Equation (1 Viewer)

[askit]

Where do you get the intuition to make x=sint and y=cost for clockwise circles?

 

[liamkk112]

View attachment 43108
Where do you get the intuition to make x=sint and y=cost for clockwise circles?

well if x =sint and y = cost at t = pi/2 u get x = 1, y = 0, at t = pi u get x = 0, y = -1, so it seems like we are going clockwise. u also know that an equation of a circle in parametrics is of some form alike to x = rcos/sin, y = rcos/sin so u always guess something of that form. additionally, we can see here that r = 2, since the magnitude of the distance from the origin to the point (sqrt3,1) is 2. also, we need to get that at t = 0, x =sqrt 3 and y = 1. if we guess x =2 sin(t-alpha) and y = 2cos(t-alpha), we can find that alpha = pi/3. so our paramterics in this case would be x= 2sin(t-pi/3) and y = 2cos(t-pi/3).

another way to get the intuition is this: plot the graph of x against sint. we can notice that from 0<x<pi that x is positive, and from pi<x<2pi x is negative. immediately this suggests that if we let x =sint that we will begin in quadrants 1 and 4 and end up in quadrants 2 and 3 in one 2pi rotation. similar thingy for y = cost, we see that from 0<x<pi/2 that x is positive, from pi/2<x<3pi/2 x is negative and that 3pi/2<x<2pi is positive. combined with the information from x =sint, this suggests that we start in the first quadrant, move into the 4th quadrant, then to the 3rd quadrant and to the 2nd in a 2pi rotation, which is clockwise movement.

 

[askit]

Would you do a table of values for most ugly 3d vector graphs like these, cause I can't really find another efficent way to go about it. I just feel like in the exam a table of values would take way to long to do accurately

 

[liamkk112]

Would you do a table of values for most ugly 3d vector graphs like these, cause I can't really find another efficent way to go about it. I just feel like in the exam a table of values would take way to long to do accurately
View attachment 43111

yeah thats an option. but slightly more efficient way:
- consider just the xy plane (basically ignore the z for now)
- figure out what is being traced out, usually it is a circle or a line-> find the direction of what's being traced out (anticlockwise/clockwise/intercepts as well for line, keep the radius/intercepts in mind)
- now just look at the z, figure out what will happen there
- if it's sine or cosine then the graph is going to oscillate up and down in z value (duh, also in this case it's very likely that one of x/y is of the form y/x=t), otherwise if it's a line (of the formz =t) then you know the graph is going to be centered around the z-axis (provided x and y are cos and sin)

in summary, what we figured out is this (for the most typical cases u will be presented with in hsc):
- z = t form +x/y=sin/cos means a helix/spiral shape, centered around the z-axis (if they're mean it might be shifted)\
- z =sin/cos form + one of x/y in t form and the other is sin/cos means a helix/spiral shape centered around whatever axis has the t form
- z,x,y all in t form means it will be a line, in this case just solving for the parametric vector equation will be easiest
-z,x,y all in sin/cos form means a weird circle shape will be traced out, kind of on an angle usually (in this case, it's usually worthwhile to do a table of values coz things will be periodic, however this is also the one case they prob won't give u)

honestly in hsc its unlikely they will ask u to graph these, it's more a multiple choice thing, so in the rare instance they do then table of values always works to figure out what's happening. internals is obv a different story though