Finding angle of rotation. Test this Monday. I missed school. Seriously need help.

Feb 2010
3
0
Ok. Here is what i do know:
in an equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0
i need to eliminate the Bxy part, and find an angle of rotation.
cot(2Ɵ) =(a-c)/b
from there i can find Ɵ and create a new x',y' equation without a Bxy part.

Here is my question:
when there are 2 possible values for Ɵ, which do i use?
this has come up on my homework a lot.

Here's an example:
xy+1 = 0
cot(2Ɵ) = 0
2Ɵ could either be (pi/2) or (3pi/2)
Ɵ could either be (pi/4) or (3pi/4)

another example:
2x² - 3xy - 2y² + 10 = 0
cot(2Ɵ) = (-4/3)
by constructing a right triangle i can figure that cos(2Ɵ) can either be (-4/5) or (4/5)

from what i can tell, sometimes either value of Ɵ will result in me making the same graph, but the x'y' equation still differs. Are both answers always equally correct? If so, then which way is the general way expected on a math test, because some answers wont ask for a graph, just the x'y' equation.

what do i do and why?
My class is a very high level honors class, and i am not going to be able to get a chance to ask the teacher before the test on monday. Thanks so much in advance.
 
Last edited:

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Ok. Here is what i do know:
in an equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0
i need to eliminate the Bxy part, and find an angle of rotation.
cot(2Ɵ) =(a-c)/b
from there i can find Ɵ and create a new x',y' equation without a Bxy part.

Here is my question:
when there are 2 possible values for Ɵ, which do i use?
this has come up on my homework a lot.

Here's an example:
xy+1 = 0
cot(2Ɵ) = 0
2Ɵ could either be (pi/2) or (3pi/2)
Ɵ could either be (pi/4) or (3pi/4)

another example:
2x² - 3xy - 2y² + 10 = 0
cot(2Ɵ) = (-4/3)
by constructing a right triangle i can figure that cos(2Ɵ) can either be (-4/5) or (4/5)
Didn't you just say -4/3? Did you mean -4/3 and 4/3?

from what i can tell, sometimes either value of Ɵ will result in me making the same graph, but the x'y' equation still differs. Are both answers always equally correct? If so, then which way is the general way expected on a math test, because some answers wont ask for a graph, just the x'y' equation.
The difference is whether you are rotating clockwise or counterclockwise and that will result in swapping x' and y' axes.

[/quote]what do i do and why?
My class is a very high level honors class, and i am not going to be able to get a chance to ask the teacher before the test on monday. Thanks so much in advance.[/QUOTE]
 
Feb 2010
3
0
thanks for the reply, but unfortunately it didnt really help.

Didn't you just say -4/3? Did you mean -4/3 and 4/3?
no, i meant what i said:
cot(2Ɵ) = (-4/3)
cos(2Ɵ) can either be (-4/5) or (4/5)
check the attached image for clarification

The difference is whether you are rotating clockwise or counterclockwise and that will result in swapping x' and y' axes.
if i choose cos(2Ɵ) = (-4/5) that puts 2Ɵ into the second quadrant and Ɵ into the first

if i choose cos(2Ɵ) = (4/5) that puts 2Ɵ into the fourth quadrant and Ɵ into the second

neither involves Ɵ being negative therefore both are standard counterclockwise rotations

please correct me if i am wrong

since both involve standard counterclockwise rotations, which way is correct?


also, in case i ever do have to graph a rotation involving a negative angle, what did you mean by swapping the x' and y' axes

thanks
 

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