The arcs in the photo at the right appear to be paths rotating about the North Star. To produce this effect, the photographer set a camera on a tripod and left the shutter open for an extended time. If the photographer left the shutter open for a full 24 hours, each arc would be a complete circle. You can model a star's "rotation" in the coordinate plane. Place the North Star at the origin. Let P(1,0) be the position of the star at the moment the camera's shutter opens. Suppose the shutter is left open for 2 hr. 40 min, with the arc ending at P'.
c. What translation rule maps P onto P"?

Question

WesternSparrows · Answer

attachment

WesternSparrows · Answer

I already know that the angle of rotation maps P onto P' is 40 degrees and that the x and y coordinates of P' are (cos 40, sin 40)

WesternSparrows · Answer

This was the photo they gave me as well.

SmokeyBrown · Answer

@westernsparrows wrote:

I already know that the angle of rotation maps P onto P' is 40 degrees and that the x and y coordinates of P' are (cos 40, sin 40)

That's good. Now, the next step would be to convert (cos 40) and (sin 40) into a coordinate in terms of unit values, probably with the help of a calculator if you have access to one or a table of values if you don't. Then, to find the translation from the original coordinate (1,0) to the new one (cos 40, sin 40), would simply be a matter of finding the difference. That way you would know exactly how many units to move up and how many units to move left, to get from the starting point to the final point.

WesternSparrows · Answer

Ok, when you say that I need to find the difference, what do you mean exactly?

WesternSparrows · Answer

so the coordinates would be: (0.67, 0.7)?

SmokeyBrown · Answer

Sure, to give another example, say that we let the camera wait for 6 hours, and the point changed from initial position (1,0) to position (0,1). The difference between Final point (0,1) and Initial point (1,0)
(xF - xI, yF-yI) or 
(0-1, 1-0) or
(-1, 1)
Just a roundabout way of saying we move to the left by 1 and move up by 1 to get from the initial point to the final point

SmokeyBrown · Answer

@westernsparrows wrote:

so the coordinates would be: (0.67, 0.7)?

The example I wrote is actually confusingly written, I think. In this case, it's simple to see that we move the point to the left by (1-0.67) =0.33 and we move it up by (0.7) units, based on that coordinate you found for the final point

WesternSparrows · Answer

Oh, okay.  That part makes sense, but what does the
(xF - xI, yF-yI) mean?

axie · Answer

@smokeybrown

SmokeyBrown · Answer

@westernsparrows wrote:

Oh, okay. That part makes sense, but what does the (xF - xI, yF-yI) mean?

Oh my bad, that's just my janky notation for "Final x-coordinate", "Initial x-coordinate", "Final y-coordinate", "Initial y-coordinate". I think I made things needlessly complicated, a plain-English explanation would have worked better in this case, haha

WesternSparrows · Answer

nope thax for ur help anyways. It makes much more sense than before:)

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