Deriving parametric equations of a point for the involute of a circle

[JoeSabs]

Homework Statement

If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end P traces an involute of the circle. In the accompanying figure, the circle in question is the circle (x^2)+(y^2)=1 and the tracing point starts at (1,0). The unwound portion of the string is tangent to the circle at Q, and t is the radian measure of the angle from the positive x-axis to segment OQ. Derive the parametric equations

x=cost+tsint, y=sint-tcost, t>0

of the point P(x,y) for the involute.

Homework Equations

?

The Attempt at a Solution

I have no idea how to do this problem! The section it's in is "Arc length and the unit-Tangent vector," but the only things explained in the section are arc length and unit tangent vector! I don't see how this relates... If anyone can provide a detailed explanation, I'd be grateful.

 

[Astronuc]

The OP is dated Sep 30, 2008

It would help to have an image.
https://mathshistory.st-andrews.ac.uk/Curves/Involute/

If one were to draw a line with one end on the circle, at some angle t, and the other at some length perpendicular to the radius at the tangent point, then one should understand what is being ask with respect to the locus of the end of the unwrapped/unwound string. One is looking for the coordinates (x,y) of the end of the string. As t increases, the length of the string increases by the angle (t) times the radius (1 for a unit circle). And one can then write (x,y) = (x(t), y(t)), where x = cos(t) + t sin(t), y = sin(t) - t cos(t).

The length of an arc is just the product of the angle subtended by the arc and the radius. The circumference of a circle is 2πr, for a semicircle, the length of the arc is 1/2 of a full circle, or πr, and so on.