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MOTION IN A CIRCLE Consider the parametric equations x = r cos t and y = r sin t These equations describe the position of a point (x,y) at
any time, t. They can be transposed into a single equation by
squaring both sides of each equation to obtain
and adding
Rearranging, we have
but
so that
which is. the equation of a circle. This means that if various values were assigned to t and the
corresponding values of x and y were calculated and plotted, the result would
be a circle. In other words, the point (x,y) moves in a circular path. Using this example again, that is and x = r cos t y = r sin t and given that
and
we are able to express the slope at any point on the
circle in terms of t. NOTE: We may find the expressions for
and
by using
calculus, but we will accept them for the present without proof. If we know
and
, we may find
which is
the slope of a curve at any point. That is,
By substituting, we find
In terms of a parameter, we see that m = cot t while in terms of rectangular coordinates, we know from
trigonometry that
