Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter3.4r

Text File | 1995-04-09 | 7KB | 183 lines

 181
à 3.4ïSecant and Cosecant Functions
äPlease find the period, phase shift, and vertical transla-
êëtion of the following functions.
â
ê Find the period, phase shift, and vertical translation of
êêêïyï=ï2∙sec 4(x + π/8) - 2

èThe period is π/2, the phase shift is π/8 units to the left, and the
vertical translation is 2 units down. (Please see Details)
éSïSince the secant is defined to be r/x, it can be expressed as
r/x = 1/(x/r) = 1/cos Θ.ïThus, sec xï=ï1/cos x.ïWe can once again
return to the Key Feature to see how to generate the basic secant func-
tion.ïSince the cos x is in the denominator, the secant function ap-
proaches infinity whenever the cos x approaches 0.ïThus, the secant
function approaches infinity at multiples of π/2.ïSince the secant is
the reciprocal of the cosine, values of the secant will be large where
values of the cosine are small, and small where values of the cosine are
large.ïThe graph of the basic secant function is given in the following
diagram.êêêêêêè y = sec Θ
@fig3401.bmp,335,195
è Like the cosine curve, the secant is
periodic with period 2π.ïIn general the
period of the curve
#yï=ïA∙sec B(x + C) + D is 2π/│B│.
The phase shift is C and the vertical
translation is D.
è You are encouraged to go to the Function
Plotter at this time to experiment with
graphs of the secant function using different
values of the letters A, B, C, and D.
 1è Find the period and phase shift of y = 1/2∙sec (x - π/16).


èA)ïThe period is π/2, and the phase shift is π/16 units to the left.
èB)ïThe period is 2π, and the phase shift is π/16 units to the right.
èC)ïThe period is π, and the phase shift is π/16 units to the right.
èD)ïå of ç
ü
èThe period is 2π, and the phase shift is π/16 units to the right.
Ç B
 2è Find the period and phase shift of y = -2∙sec (2x + π/3).


èA)ïThe period is π/2, and the phase shift is -2 units to the left.
èB)ïThe period is 2π, and the phase shift is π/3 units to the left.
èC)ïThe period is π, and the phase shift is π/6 units to the left.
èD)ïå of ç
ü
èThe period is π, and the phase shift is π/6 units to the left.
Ç C
 3è Find the period and phase shift of y = sec π(x - 1/8).


èA)ïThe period is 2, and the phase shift is 1/8 units to the right.
èB)ïThe period is π, and the phase shift is π/8 units to the right.
èC)ïThe period is π/2, and the phase shift is π/8 units to the left.
èD)ïå of ç
ü
èThe period is 2, and the phase shift is 1/8 units to the right.
Ç A
äPlease find the period, phase shift, and vertical transla-
êëtion of the following functions.
â
ê Find the period, phase shift, and vertical translation of
êêêïyï=ï2∙csc 4(x + π/8) - 2

èThe period is π/2, the phase shift is π/8 units to the left, and the
vertical translation is 2 units down. (Please see Details)
éSïSince the cosecant is defined to be r/y, it can be expressed as
r/y = 1/(y/r) = 1/sin Θ.ïThus, sec xï=ï1/sin x.ïWe can once again
return to the Key Feature to see how to generate the basic cosecant func-
tion.ïSince the sin x is in the denominator, the cosecant function ap-
proaches infinity whenever the sin x approaches 0.ïThus, the cosecant
function approaches infinity at multiples of π.ïSince the cosecant is
the reciprocal of the sine, values of the cosecant will be large where
values of the sine are small, and small where values of the sine are
large.ïThe graph of the basic cosecant function is given in the follow-
ing diagram.êêêêêêy = csc Θ
@fig3402.bmp,292,195
è Like the sine curve, the cosecant
is periodic with period 2π.ïIn general
the period of the curve
# yï=ïA∙csc B(x + C) + D is 2π/│B│.
The phase shift is C and the vertical
translation is D.
è You are encouraged to go to the
Function Plotter at this time to ex-
periment with graphs of the cosecant
function using different values of
the letters A, B, C, and D.
 4è Find the period and phase shift of y = 1/2∙csc (x - π/16).


èA)ïThe period is π/2, and the phase shift is π/16 units to the left.
èB)ïThe period is 2π, and the phase shift is π/16 units to the right.
èC)ïThe period is π, and the phase shift is π/16 units to the right.
èD)ïå of ç
ü
èThe period is 2π, and the phase shift is π/16 units to the right.
Ç B
 5è Find the period and phase shift of y = -2∙csc (2x + π/3).


èA)ïThe period is π/2, and the phase shift is -2 units to the left.
èB)ïThe period is 2π, and the phase shift is π/3 units to the left.
èC)ïThe period is π, and the phase shift is π/6 units to the left.
èD)ïå of ç
ü
èThe period is π, and the phase shift is π/6 units to the left.
Ç C
 6è Find the period and phase shift of y = csc π(x - 1/8).


èA)ïThe period is 2, and the phase shift is 1/8 units to the right.
èB)ïThe period is π, and the phase shift is π/8 units to the right.
èC)ïThe period is π/2, and the phase shift is π/8 units to the left.
èD)ïå of ç
ü
èThe period is 2, and the phase shift is 1/8 units to the right.
Ç A
äPlease graph one period of the following functions.

â
êêêëGraph one period of
êêêëyï=ïsec 4(x + π/8)

êêè (Please see Details for an explanation.)
éSïYou can graph yï=ïsec 4(x + π/8) on the Function Plotter in
this program.ïYou are encouraged to go to the Function Plotter at this
time and graph this function.
è It is also possible to graph this functions by hand using the reci-
procal relation, sec xï=ï1/cos x.ïFirst, draw a dotted graph of the
reciprocal function, yï=ïcos 4(x + π/8).ïThe period is π/2, and the
phase shift is π/8 units to the left.

@fig3403.bmp,25,135
êêè Then, draw vertical asymptotes where
êêè the cosine is zero.ïFinally, draw
êêè the secant function thinking of the
êêè reciprocal relationship.
@fig3404.bmp,425,125



è The solid curve is a graph of one period of y = sec 4(x + π/8).ïYou
can also graph the cosecant function by hand by first drawing a dotted
version of the corresponding reciprocal sine function, then using the
reciprocal relationship to graph the cosecant.
 7

èGraph one period ofë A)êêêïB)
ïyï=ï1/2∙sec (x - π/16).

@fig3405.bmp,125,230
@fig3406.bmp,125,230
ü
êêêêThe period is 2π, the phase shift
êêêêis π/16 units to the right, and
êêêêthe reciprocal function is
êêêêïyï=ï1/2∙cos (x - π/16)


@fig3405.bmp,125,245
Ç A
 8

è Graph one period ofëA)êêêïB)
è yï=ï-2∙csc (2x + π/3).

@fig3407.bmp,125,230
@fig3408.bmp,125,230
üêêè The period is π, the phase shift is π/6
êêêèunits to the left, and
êêêèthe reciprocal function is
êêêêyï=ï-2∙cos 2(x + π/6).



@fig3408.bmp,125,245
Ç B