SOLUTION: A circular point P(a,b) moves clockwise around the circumference of a unit circle starting at (1,0) and stops after covering a distance of 8.305 units. Find the coordinates of wher

Algebra ->  Trigonometry-basics -> SOLUTION: A circular point P(a,b) moves clockwise around the circumference of a unit circle starting at (1,0) and stops after covering a distance of 8.305 units. Find the coordinates of wher      Log On


   

Question 1077717: A circular point P(a,b) moves clockwise around the circumference of a unit circle starting at (1,0) and stops after covering a distance of 8.305 units. Find the coordinates of where P stops (round to three decimal places) and what quadrant it lies in.
Can someone help me with this?
Thank you.
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
.
A highlight%28cross%28circular%29%29 point P(a,b) moves clockwise around the circumference of a unit circle starting at (1,0)
and stops after covering a distance of 8.305 units. Find the coordinates of where P stops (round to three decimal places)
and what quadrant it lies in.
Can someone help me with this?
Thank you.
~~~~~~~~~~~~~~~~~~ 
8.305+-+2%2Api%2Ar = 8.305+-+2%2A3.1416%2A1 = 8.305+-+6.283 = 2.022.


The point makes 1 full revolution clockwise and stops in QIII at the terminate angle alpha = -2.022 radians.

QIII, third quadrant.


Notice: the point does not have a shape.

Therefore, do not say "circular point".


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The tutor "KMST" (whose contribution I highly estime) missed the words "the point P moves clockwise".



Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
red%28EDITED%29 (corrected based on misreading pointed out by ikleyn)
Thanks ikleyn.

The point traveled red%28clockwise%29 8.305 times the radius, and the radius of the unit circle is 1.
That means the point swept and angle of cross%288.305%29%29red%28-8.305%29%29radians .
Its coordinates are
cross%28x=cos%288.305%29=about%29cross%28highlight%28-0.436%29%29
x=cos%28red%28-8.305%29%29=abouthighlight%28-0.436%29%29, and
cross%28y=sin%288.305%29=about%29cross%28highlight%280.900%29%29
y=sin%28red%28-8.305%29%29=abouthighlight%28red%28-0.900%29%29 .
Those coordinates tell you the point is in quadrant cross%28II%29 red%28III%29 ,
because x is negative and y is cross%28positive%29 red%28negative%29 .

For another point of view:
Since 2pi=about6.283 is one turn red%28clockwise%29 around the circle,
the point is now 8.305-6.283=2.022 radians into its second lap around the circle.
That is less than half a turn, which would be pi=3.1416 radians,
so it has to be in quadrant cross%28I%29 red%28IV%29 or cross%28II%29 red%28III%29 .
red%28One%29red%28quadrant%29red%28amounts%29red%28to%29red%28pi%2F2=1.571%29 red%28%22radians+%3B%22%29
red%28two%29red%28quadrants%29red%28amount%29red%28to%29red%28pi=3.142%29 red%28%22radians+%2C%22%29
red%28so%29red%28P%29red%28is%29red%28in%29red%28quadrant%29red%28III%29 .

You could also divide 8.305 by pi%2F2=1.571
to see that the point covered 5 whole quadrants,
and part of a 6th quadrant,
so with the first 4 quadrants being one whole turn,
started a second lap, and it is now in the second quadrant
red%28of%29 red%28its%29 red%28second%29 red%28clockwise%29 red%28lap%29 .
red%28%22So%2C%22%29red%28P%29red%28is%29red%28in%29red%28quadrant%29red%28III%29 .