1. Ant on Turntable

" A turntable is spinning in the xy plane in a counterclockwise direction with a radius of 1 unit at a rotational velocity of 1 radian per second. A chord is drawn from (1,0) to (0,1) and an ant is placed at (1,0). The table begins to turn and the ant begins to walk at 1 unit per second with respect to the point he starts at.

Find a parametric equation that gives the position of the ant at time t from t=0 to when the ant reaches the end of the chord. "

I tried to write an equation to find the ant's position relative to the starting point, so I integrated the vector that describes the ant's direction. Then I added the vector that describes the position of the starting point. I got
R = cos(t)+sin(3PI/4+t)-sin(3PI/4) i + sin(t) - cos(3PI/4+t)+cos(3PI/4)

Unfortunately my book gives a much different answer

Can anyone tell me what I did wrong?
Thanks

2. The way I see it, if the turntable stayed stationary then at time t seconds the ant would reach the point $\displaystyle \Bigl(1-\frac t{\sqrt2},\frac t{\sqrt2}\Bigr)$ on the line x+y=1. But in that time the turntable will actually have rotated through an angle t radians, so the ant will be at the point $\displaystyle \Bigl(\Bigl(1-\frac t{\sqrt2}\Bigr)\cos t - \frac t{\sqrt2}\sin t, \Bigl(1-\frac t{\sqrt2}\Bigr)\sin t + \frac t{\sqrt2}\cos t\Bigr)$. It reaches the end of the chord at time $\displaystyle t=\sqrt2$, when it will be at the point $\displaystyle (-\sin\sqrt2,\cos\sqrt2)$.