object 210' away is moved 6 degrees how many feet will it move?

Answer 1

sin(6)*210 = 21.9 feet

Answer 2

The question is somewhat vague in that it does not tell how the object moved the 6 degrees.
I can think of 2 separate ways this could have happened. The object could have moved some distance y perpendicular to an imaginary line which connected the observer with the object (in this case, the distance to the object is now greater than before), or the object could have moved along a circular path (in which case, the distance to the object is the same). Depending on which of these occurred, the answer will be slightly different.

Option 1)
The object moved a distance y perpendicular to the radial vector between it and the observer.
Being able to draw a diagram would help immensely, but this is what it was like before,

0------------X
Where 0 is the object, X is the observer. After it moved,
0
|_________X
In this case, the original distance to the object is not eh adjacent side of the newly formed triangle.

So solve for the distance the object moved we need to multiply the length of the hypotenuse by the sine of the angle (6 degrees). But we no longer know the length of the hypotenuse, to find this we need to divide the adjacent length by the cosine of the angle.
Hyp = adj / cos (6) = 210 / cos (6)
Hyp = 211.16 ft.
Now we multiply this by the sin (6),
Opp = y distance moved = hyp * sin (6) = 211.16 * sin (6)
Opp = 22.07 ft.
So in this case, the object moved 22.07 ft. from its original position.


Option 2)
The Object moved along a circular path such that the initial distance of the observe from the object is the same as the final distance.

0-------------X
When the object moves,
0
|________X
|
That is not the best picture, but it gets the job done.
The total distance the object moves is now equal to 2 times the sine of 1/2 the angle between the lower part of the triangle an the upper part.
The Distance moved = 210 ft. * 2 * sin (3 degrees) = 21.98 ft.
We are essentially breaking up the isosceles triangle into 2 right triangles, finding the opposite side lengths and then adding them together. The distance moved in this case is the linear, absolute, distance moved, not taking into account whether the object actually did or did not follow a circular path to get to this point.