Download Graphs of tan, cot, csc, and sec Functions

Graphs of tan, cot, csc, and sec Functions:
09/19/2012
Goal:
# Plot graph of tan, cot, sec and cosecant.
# Graph of the form y = Atan(ωx) + B, y = cot(ωx) + B
# Graph of the form y = Acsc(ωx) + B, y = Asec(ωx) + B
The Graph of tan function :
Because the tangent function has period π, we only need to determine the
graph over some interval of length π. The rest of the graph will consist of rep, −π
, π , 3π , ...
etitions of that graph. The tangent function is not defined at ..., −3π
2
2 2 2
, π ), of length π. We find points in
we can concentrate on the interval ( −π
2 2
−π π
interval ( 2 , 2 ) and plot the points and connect them with a smooth curve.
Class discussion: plot the graph for tan
Graph of y = Atan(ωx) + B :
For tangent function, since range is all real number so there is no concept
of amplitude. The role of A in y = Atan(ωx) + B provide the magnitude of
vertical stretch. The period of tanx is π so period of y = Atan(ωx) + B is
π
, caused by horizontal compression of the graph by a factor of ω1 . Finally,
ω
the presence of B indicates that a vertical shift is required.
class discussion: plot graph of y = 2tan(ωx) − 1
NOTE: Plot cot and y = Acot(ωx) + B as for tangent case above.
Graph of csc and sec :
The cosecant and secant functions, sometimes referred to as reciporcal
functions, are graphed by making use of the reciprocal identities
1
1
cscx = sinx
and secx = cosx
1
Example: The value cosecant function y = cscx at a given number x
equals the reciprocal of the corresponding value of the sine function, provided that the value of sine function is not 0. If the value of sinx is 0, then
x is an integer multiple of π. At such numbers, the cosecant function is not
defined. In fact, the graph of the cosecant function has vertical asymptotes
at integer multiples of π.
Class discussion : plot graph of cosecant.
Using same idea as above we can obtain the graph of y = secx.
Graph of y = Acsc(wx) + B :
The role of A is to set range. The range of y = Acsc(wx) + B is {y||y| > A},
due to vertical stretch of the graph by a factor of |A|. Period of Acsc(wx)
, due to the horizontal compression of the graph by a factor of w1 . The
= 2π
w
presence of B indicates that a vertical shift is required.
NOTE: We can plot graph of y = sec(wx) + B as y = Acsc(wx) + B.
Class discussion: Plot graph of y = Acsc(wx) + B
.... ....
2