# Thread: Applied Algebra x,y coordinates please help

1. ## Applied Algebra x,y coordinates please help

It reads: When you rotate the x-y coordinate system by an angle of $\theta$, the new cordinates x' and y' can be calculated using the following expressions:

x'= xCos $\theta$+ ysin $\theta$ (1)
y'=-xsin $\theta$ + ycos $\theta$(2)

Given that you know the rotated coordinates x' and y' and the rotation angle $\theta$ how do you calculate the original x an y? (in other words, how do you express x and y using x', y' and $\theta$?

Hint: rearrange equations (1) and (2) so that x and y can be expressed using x', y', sin $\theta$ and cos $\theta$

and thats all it gives me except for the addition of another example problem which asks

if x'=24, y'=-3 and $\theta$=30 degrees, what are the values of x and y?

many thanks, also what are these problems trying to show im not really sure what they have to do with college algebra.

2. It is just reversing the procedure.

To get the (x',y') from (x,y) with a rotation of theta, the following were derived:
x'= xCos(t) + ysin(t) --------(1)
y'= -xsin(t) + ycos(t) ..........(2)

So if you reverse the procedure, to get back to the (x,y), you use
x = x'Cos(-t) + y'sin(-t) --------(3)
y'= -x'sin(-t) + y'cos(-t) ..........(4)
where the angle of rotation is the reverse of the theta before.

Rewriting those, in terms of the original theta,
x = x'cos(t) -y'sint(t) -------(3a)
y = x'sin(t) +y'cos(t) --------(4a)

that is because cos(-t) = cos(t), and sin(-t) = -sin(t).

--------------------------
'"if x'=24, y'=-3 and =30 degrees, what are the values of x and y?"

x = 24cos(30deg) -(-3)sin(30deg) = 22.28461
y = 24sin(30deg) +(-3)cos(30deg) = 9.40192

Maybe you want to check if (22.28461,9.40192) will become (24,-3) if the original axes are rotated by 30 degrees.

3. Thank you sir! It is crystal clear now