Unformatted text preview: all real numbers t. Then, in particular,
cos(0 + p) = cos(0) so that cos(p) = 1. From this we know p is a multiple of 2π and, since the
smallest positive multiple of 2π is 2π itself, we have the result. Similarly, we can show g (t) = sin(t)
is also periodic with 2π as its period.2 Having period 2π essentially means that we can completely
understand everything about the functions f (t) = cos(t) and g (t) = sin(t) by studying one interval
of length 2π , say [0, 2π ].3
One last property of the functions f (t) = cos(t) and g (t) = sin(t) is worth pointing out: both of
these functions are continuous and smooth. Recall from Section 3.1 that geometrically this means
the graphs of the cosine and sine functions have no jumps, gaps, holes in the graph, asymptotes,
corners or cusps. As we shall see, the graphs of both f (t) = cos(t) and g (t) = sin(t) meander nicely
and don’t cause any trouble. We summarize these facts in the following theorem.
1 See section 1.7 for a review of these concepts.
Alternatively, we can use the Cofunction Identities in Theorem 10.14 to show that g (t) = sin(t) is pe...
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