Alan G. answered • 06/01/16
Tutor
5
(4)
Successful at helping students improve in math!
The best route to solving this is to use parametric equations for the circle.
The Cartesian equation of the circle is
(x - 3)2 + (y - 2)2 = 1.
Let cos θ = x - 3 and sin θ = y - 2. After you solve for x and y, you get the parametric equations
x = cos θ + 3, y = sin θ + 2. In these equations, θ represents the counterclockwise angle of rotation, with θ = 0 putting the point at (4,2).
The coordinates of the rotated point are then (x,y), where x and y are as stated above.
Let me know if you require further explanation.
Alan G.
Moxa,
Ok, here is more explanation.
The equation of the circle is found from the center and radius, which were given.
The parametric equations were formed from the trigonometric identity cos2 θ + sin2 θ = 1 and substitution. This is a common technique for finding parametric equations. Another way to see this is to use the equation of the unit circle, x2 + y2 = 1, and replace x = cos θ and y = sin θ. By plugging in angles for θ such as 0, π/2, π, 3π/2, etc., you can verify that the angle θ represents the angle of rotation (in radians) from the point (4,2) in a counterclockwise direction.
The final answer is just the result of performing algebra on these two equations and solving for the variables x and y.
Since your original question assumes the angle θ is in degrees, you can tweak the answer I gave the first time by replacing θ by 180θ/π. This will make the use of θ in degrees possible.
Thus, the final form of the answer would be
x = 3 + cos (180θ/π), y = 2 + sin (180θ/π).
(I apologize for stating the answer without the 180/π conversion factor. I am so used to doing this in radian measure that I assumed that was what you were looking for.)
Please reply one more time if you need any further clarification.
Report
06/02/16
Amy H.
How would I divide that
Report
06/02/16
Alan G.
Moxa,
Divide what? Actually, if you start with the angle θ in degrees, multiplying it by π/180 will convert it to radians. This means that the expressions in the parentheses are really incorrect. You should replace them by (π/180)θ.
Here is another try (hopefully my last):
x = 3 + cos (π/180)θ, y = 2 + sin (π/180)θ.
Have I answered your question?
Report
06/02/16
Amy H.
Yea sorry about that
Report
06/02/16
Alan G.
Okay, that is great!
Report
06/02/16
Hazel H.
im still confused why its π/1800 for θ
Report
06/12/23
Amy H.
06/01/16