" 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