Stitz-Zeager_College_Algebra_e-book

# 3 sin3 x sin2 x 3 sin3 x sin2 x 0 sin2 x3 sinx

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Unformatted text preview: (x) tan(π ) 1 − (tan(x))(0) which tells us the period of tan(x) is at most π . To show that it is exactly π , suppose p is a positive real number so that tan(x + p) = tan(x) for all real numbers x. For x = 0, we have tan(p) = tan(0 + p) = tan(0) = 0, which means p is a multiple of π . The smallest positive multiple of π is π itself, so we have established the result. We take as our fundamental cycle for y = tan(x) the interval − π , π , and use as our ‘quarter marks’ x = − π , − π , 0, π and π . From the graph, we 22 2 4 4 2 see conﬁrmation of our domain and range work in Section 10.3.1. It should be no surprise that K (x) = cot(x) behaves similarly to J (x) = tan(x). Plotting cot(x) over the interval [0, 2π ] results in the graph below. y x 0 cot(x) (x, cot(x)) undeﬁned π 4 π 2 3π 4 0 π 4,1 π 2,0 −1 3π 4 , −1 π undeﬁned 5π 4 3π 2 7π 4 1 −1 2π undeﬁned 1 0 5π 4 ,1 3π 2 ,0 7π 4 , −1 1 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 2π x −1 The graph of y = cot(x) over [0, 2π ]. From these data, it clear...
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