Finding coordinates on a circle

Question

So this problem I am have difficulty with. I think where I am going wrong is how to calculate the initial theta. Do I just use pi/2 because in the pictures it show to angle theta off the 90 degree axis?

Answer 1

Well, after $4$ minutes, your merry-go-round would have revolved $9.2$ times, and been displaced from its original position $0.2 \cdot 360 = 72^\circ$. To begin with, $\theta = 33^\circ$, but after $4$ minutes, its new value would be $33 + 72 = 105^\circ$. It is given that the merry-go-round has a radius of $22$ feet, so putting it all together, your new coordinates after $4$ minutes would be $22\angle 105^\circ$ in polar form. $22\cos(105) \approx -5.694$ and $22 \sin(105) \approx 21.250$, giving you a final answer of $\displaystyle{(x,y) = (-5.694,21.250)}$

(This was only the answer to the first problem, but I'm sure you can do the others using similar logic.)

Answer 2

(1) decide whether you are going to work in degrees or radians. Since everything in the problem is in degrees I'm going for degrees (otherwise you have to convert everything to radians).

(2) The initial theta is stated in the problem. For the one you show as wrong, theta = 33 degrees.

(3) after four minutes the merry-go-round has rotated 2.3 * 4 times = 9.2 times.

(4) each complete rotation leaves the point in the same place, so it is just the .2 that makes a difference.

(5) .2 of a rotation is 0.2 * 360 = 72 degrees.

(6) add this to the initial theta of 33 degrees to see that the effective rotation is 105 degrees.

(7) as you had, use cosines and sines to compute x and y, but with the correct angle (x, y) = (22 cos 105, 22 sin 105). Look up the values and calculate.