theta=sin^-1 (5/13) ; (x,y)=(-3,1)
how do u find the coordinates of x' and y'??
what is (x',y')? (x,y) rotated around (0,0) through the angle theta?
yes
the x' y' is x pime y prime
ok, so just plug in your numbers:
x' = x cosθ + y sinθ
y' = -x sinθ + y cosθ
If sinθ = 5/13, then cosθ = 12/13
i thought the origianl formula is
x = x' cosθ + y' sinθ
y = x' sinθ - y' cosθ
It is. But you already have (x,y) and want (x',y')
Your formula gives you (x,y) if you have (x',y'). If you solve it for x,y you will get my formula.
In fact, since (x',y') is (x,y) rotated through θ , that makes (x,y) the image of (x',y') rotated through -θ , which is your formula.
okk i have a similar Q.
θ=45degree ; (x,y)=(0, -2)
i got to 0=sqrt2/2 (x'-y')
-2=sqrt2/2 (x'+ y')
then i got stuck....
since sinθ = cosθ = 1/√2,
x' = x/√2 + y/√2 = 1/√2(0-2) = -2/√2 = -√2
y' = -x/√2 + y/√2 = 1/√2 (0-2) = -2/√2 = -√2
so, (x',y') = (-√2,-√2)
Hmmm. It appears that my formula was in error, since clearly we want to end up with (√2,-√2)
So, I must have mixed up my + and - signs. Your formula may be correct after all.
One of us needs to review, and since it's your grade, it might as well be you. :-( Choose points that make it easy to see what your destination must be, and go for it.
come on help mee~~
To find the coordinates of x' and y', we first need to understand what theta represents in this context. Theta is the angle whose sine value is 5/13.
Given that theta = sin^-1(5/13), we can use this information to find x' and y'.
Recall that the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. Therefore, we can create a right triangle where the opposite side is 5 and the hypotenuse is 13.
Now, let's consider the coordinates of point P(-3, 1). This point represents a specific position in a coordinate plane.
To find the coordinates of x' and y', we need to rotate point P by an angle theta counterclockwise around the origin (0,0).
To do this, we can use the following rotation formulas:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
In our case, x = -3, y = 1, and theta = sin^-1(5/13). Plugging in these values, we can calculate x' and y':
x' = -3 * cos(sin^-1(5/13)) - 1 * sin(sin^-1(5/13))
y' = -3 * sin(sin^-1(5/13)) + 1 * cos(sin^-1(5/13))
To compute the values of x' and y', we can substitute sin^-1(5/13) with its numerical value, approximately 0.3948.
x' = -3 * cos(0.3948) - 1 * sin(0.3948)
y' = -3 * sin(0.3948) + 1 * cos(0.3948)
After evaluating these equations, you will find the coordinates of x' and y' which represent the rotated position of point P.