Help! I don't understand this problem
Find the angle of rotation that maps point P (1,0) on to P'. What are the x and y coordinates of point P' to the nearest thousandths? Determine a translation rule that maps point P onto P'.
gotta know what P' is, dontcha think?
I will say that a rotation of angle θ maps
(0,1) -> (cosθ,sinθ)
sorry - that's (1,0)
To find the angle of rotation that maps point P (1,0) onto point P', we need to first determine the coordinates of P'.
To do that, we can imagine a unit circle centered at the origin (0,0) with point P (1,0) lying on the positive x-axis. If we rotate the point P counterclockwise around the origin by some angle, it will map to a new point P'.
Since the x-coordinate of P is 1, and the y-coordinate is 0, we can use trigonometric functions to determine the coordinates of P' after rotation.
The x-coordinate of P' can be found using the cosine function. The cosine of an angle is equal to the adjacent side divided by the hypotenuse. In this case, the adjacent side length is the x-coordinate of P' (let's call it x'), and the hypotenuse length is the radius of the unit circle, which is 1. So we have:
cos(angle) = x' / 1
Simplifying, we get:
cos(angle) = x'
Now, we need to find the y-coordinate of P'. The y-coordinate can be found using the sine function. The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side length is the y-coordinate of P' (let's call it y'), and the hypotenuse length is 1. So we have:
sin(angle) = y' / 1
Simplifying, we get:
sin(angle) = y'
Now, we can use a trigonometric function or calculator to find the value of the angle that satisfies these equations. Let's say we find that angle to be θ.
Once we have the angle θ, we can determine the x and y coordinates of P' by substituting θ into the equations:
x' = cos(θ)
y' = sin(θ)
These values will give you the x and y coordinates of point P' to the nearest thousandths.
To determine a translation rule that maps point P onto P', we need to find the difference between the x and y coordinates of P' and P. This will give us the amounts by which P needs to be translated in order to reach P'.
The translation rule can be written in the form (x', y') = (x + a, y + b), where (x,y) are the coordinates of point P, and (x',y') are the coordinates of point P'. The values of a and b can be found by subtracting the x and y coordinates of P from the x and y coordinates of P', respectively:
a = x' - x
b = y' - y
Substituting the x and y values, as well as the previously calculated x' and y' values, will give you the translation rule that maps point P onto point P'.