Question

can u please help me on part C?
The arcs in the photo at the right appear to be paths of stars 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 h 40 min, with the arc ending at P`.
a. What angle of rotation maps P onto P`?
b. What are the x- and y-coordinates of P` to the nearest thousandth?
c. What translation rule maps P onto P`?http://assets.openstudy.com/updates/attachments/57559122e4b061baa7cbdb8e-sweetheart-1465226540364-imag2642.jpg

Answer 1

24 hours produces a full rotation of 360 degrees.
2 h and 40 min is what fraction of 24 hours?

Answer 2

160 /1440 = 10 div 10 = 16/144 = 16 div 16 = 1/9

Answer 3

@mathstudent55 it = 1/9

Answer 4

Correct.
2 h and 40 min are 1/9 of 24 hours.
If in 24 hours, a star rotates 360 deg, then in 2 h and 40 min, the start rotates 1/9 of 360 deg.
What is 1/9 of 360 deg?

Answer 5

40

Answer 6

@mathstudent55 is it 40

Answer 7

Correct.
That is the answer to part a.

Answer 8

would you put both answer

Answer 9

Now for part b., you need the coordinates of point P'.

Answer 10

Since point P(1, 0) is on the unit circle, when that point rotates about the origin to point P', point P' will also be on the unit circle.

Answer 11

The coordinates of any point on the unit circle are simply the cosine and the sine of the angle that the terminal side that goes through that point makes with the positive x-axis.

Answer 12

Since in this case, point P' is on a terminal side of an angle that is 40 deg from the positive x-axis. the coordinates of point P" are \((\cos 40^\circ, \sin 40^\circ)\).