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Let
G of t
equals cosecant of t. We want to discuss its domain.
Cosecant of t
is the reciprocal of sine of t. The domain
of G of t is the set of t,
t can be any real number
except,
sine of t
cannot equal to 0; otherwise the denominator
will be 0 and we will be dividing 1 by
a zero number. If we draw the x- and y-axes.
Sine of t will represent
by the y-value.
Sine of t equals 0 when y=0 and that happens
when the angle in standard position is equal to 0,
or pi,
or 2 pi, or 3 pi,
and so on. Thus,
the domain equals the set of t such that t
cannot equal to k pi
where k is any integer.
We draw a number line to see how this works.
It can be
any new number except when t =
0, that happens when k is the integer 0;
and when
k=1, t= pi;
when k=negative 1
t
equals negative pi, and so on.
So, what happens,
is that when we have k times pi,
there is a hole there. And then, the next hole
will be k plus one in parentheses, times
pi. We use the extended interval notation
to write down the domain. It equals,
a big union sign, from k
equals negative infinity to infinity,
k can be any integers, then write down a set of parenthese,
k pi, comma,
the sum k plus one,
times pi.
That's our domain.