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Unformatted text preview: 6.5Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions: T:
2 Undeﬁned y = tan x, —00 < x <1 00, x not equal to odd multiples of Properties of the Tangent Functioh ’i’T 9 1. The domain is the set of all real numbers, except. odd multiples of 2. The range is the set of: all real numbers. 3. The tangent function is an odd function, as the symmetry of the graph with
respect to the origin indicates. 4. The tangent function is periodic. with period 77. S. The yrs—intercepts are .. ., —'27r. —7r, 0. “m 217, 377‘. . .; the yintercept is 0. 6. Vertical asymptotes occur atx = .. . . ‘94 y = cot X, —0C% “a. x «a: 00,): not equal to integer multiples of in “C30 «if... y «1: DO The Graph of the cosecant function: y 2 cs»: x, —0<3 <5: x <2: 00, 9: not equal to integer multiples «of 17;. yl 3E 1 The Gra h of the secant function: y = sec X. ——C>O <1 x ~11 00,): not equal ,— to odd multiples of L, yl E 1 2 Tangent Function: .3; : tam‘7)
73' I"? One cycle occurs between 2 and 3 .
There are vertical asymptotes at each end of the cycle. 3T The asymptote that occurs at Z repeats every 4”? units.
period: 5’? amplitude: none, graphs go on forever in vertical
directions. Cotangent Function: : COW?) One cycle occurs between 0 and J? . There are vertical asymptotes at each end of the cycle.
The asymptote that occurs at 1"? repeats every 1'? units.
period: FT amplitude: none, graphs go on forever in vertical
directions . The xintercepts of the graph of y = tan(x) are the
asymptotes cf the graph of y = cot(x). The asymptotes of the graph of y = tan(x) are the x
intercepts of the graph of y = cot(x). The graphs of y = tan(x) and y = cot(x) have the same x
values for yvalues of l and 1. Note: The graphs of y = tan(x) and y = cot(x) "face" in
opposite directions. y = tan (x) E ‘ The‘graph' does'not STOP even though the
plot may "appear" as if the graph stops J
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I .L—quwh—u—L———Mw as the yvalues increase/decrease. Cosecant Function: = CSCCY) There are vertical asymptotes. The asymptote that occurs
at 1"? repeats every I"? units. period: 2»? amplitude: none, graphs go on forever in vertical
directions. The maximum values of y = sin x are minimum values of
the positive sections of y = csc x. The minimum values of
y = sin x are the maximum values of the negative sections
of y = csc x. The xintercepts of y = sin x are the asymptotes for y =
csc x. Note: the U shapes of the cosecant graph are tangent to its
reciprocal function, sine, at sine's max and min locations. Secant Function: = SeqT) There are vertical asymptotes. The asymptote that occurs
1‘? at 2 repeats every 2”? units.
period: 33"
amplitude: none, graphs go on forever in vertical
directions. The maximum values of y = cos x are minimum values of
the positive sections of y = sec x. The minimum values of
y = cos x are the maximum values of the negative sections
of y = sec x. The x—intercepts of y = cos x are the asymptotes for y = sec x. Note: the U shapes of the secant graph are tangent to its
reciprocal function, cosine, at cosine‘s max and min
locations. I!;..¢xm...1rl ‘1vn .
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 Fall '11
 BlisinHestiyas
 Mathematical analysis, Multiplicative inverse, Tangent Functioh

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