A wheel of radius one foot is placed so that its center is at the origin, and a paint spot on the rim is at (1,0).

Question

The wheel is spun 37 degrees in a counterclockw direction. What are the coordinates of the paint spot? What if the wheel is spun θ degrees instead?

Answer 1

Hi Mioxa,

 

You may as well just work this problem out for a general θ and then you can plug some specific value like 37 deg in later.

 

You can imagine a radial extending from the origin to the paint spot. Initially, the radial would be horizontal and to the right. The angle θ is measured from the positive horizontal, or x, axis, and so the initial value of θ is 0.

 

If we sweep the radial in a counter-clockwise sense so that a general angle θ is subtended, then you can always imagine the radial forming the hypotenuse of a right triangle. You can draw a vertical from the paint spot to the x axis. That vertical forms a right angle with the x axis.

 

 

     /|

    / |

 r /  |

  /   | y

 /    |

/     |

θ___

   x

 

The angle θ in my picture would be at the bottom left corner where the radial meets the origin. For the (x,y) coordinates of the paint spot at any angle θ, you can use Trigonometry.

 

cos(θ) = x/r  or x = rcos(θ)

 

sin(θ) = y/r, or y = rsin(θ)

 

In this case the radial is of size 1, so r = 1. So, for any θ we have

 

x = cos(θ)

y = sin(θ)

 

For θ = 37 deg, we have

x = cos(37 deg) = 0.8

y = sin(37 deg) = 0.6