By denition the real number t arcsin 1 satises 0 t

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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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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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