Question 756882
{{{drawing(300,300,-1.1,1.1,-1.1,1.1,
grid(0),
circle(0,0,1),
red(line(0,0,-5/13,12/13)),
locate(0.015,0.1,O),locate(1.01,0.11,A),
locate(-5/13,1.04,P),
blue(line(5/13,-12/13,0,0)),locate(0.4,-0.92,Q),
green(line(-5/13,-12/13,5/13,12/13)),
locate(-5/13,-0.92,B),locate(5/13,1.04,C)
)}}} Angle AOP, swept counterclockwise from OA to OP corresponds to {{{t}}}
Maybe {{{t}}} is the measure of AOP, or the measure of AOP plus a number of whole turns.
An angle measuring {{{2pi}}} would be one whole counterclockwise turn.
An angle measuring {{{pi}}} would be half of a counterclockwise turn.
An angle measuring {{{-pi}}} would be half of a clockwise turn.
 
i. Adding or subtracting {{{pi}}} to angle {{{t}}} would take you to point Q, halfway across the circle.
{{{P(t-pi)=Q(5/13,-12/13)}}}
Adding or subtracting {{{2pi}}} would take you a full turn back to the same spot.
{{{P(t-3pi)=P(t-pi-2pi)=P(t-pi)=highlight(Q(5/13,-12/13))}}}
 
ii. A minus sign as in {{{-t}}} means go in the opposite direction, so {{{P(-t)=B(-5/13,-12/13)}}}.
Then, adding {{{pi}}} would take you halfway around the circle to
{{{highlight(P(-t+pi)=C(5/13,12/13))}}}
 
2. If P(t)=(7/25,24/25), P(t-pi) is halfway across the circle at a point {{{X}}}
{{{highlight(P(t-pi)=X(-7/25,-24/25))}}}