# Graphs of Tangent, Cotangent, Secant, and Cosecant

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```					  Graphs of Tangent,
Cotangent, Secant, and
Cosecant

5.5
Tangent
   Tan x =
   sin x / cos x
Creating Tangent Graphs
F(x) = tan x
1. Domain       1.All reals except odd multiples
of π /2
2. Range
2.All reals
3. Continuity
3.Continuous on its domain
4. Inc & dec
4.Increasing on each interval in
its domain.
F(x) = tan x
5. Symmetry     5. Odd
6. Boundedness 6. Unbounded
7. Max/min      7. No max/min
8. Asymptotes   8. No horizontal asymptotes—
VA at k*π/2 where k is an odd
integer.
atan(b(x-h)) +k
   a yields a vertical stretch(>1) or
shrink(<1, >0).
   H causes a horizontal translation
   K causes a vertical translation
   B affects the period. Stretch(<1, >0) or
shrink horizontally(>1). Period is π/|b|.
Tangent
   Graphing tan(x):
• Find the period which is π/|b| or π/|1| or π.
• Divide the period by 4. π/4.
   Sine at 0, π/4, π/2, 3π/4, π is
•0      √2/2   1   √2/2        0
   Cosine at 0, π/4, π/2, 3π/4, π is
•1      √2/2   0   -√2/2       -1
   So tangent at 0, π/4, π/2, 3π/4, π is
•   0   1   undefined     -1       0
Cotangent
   cot x =
   1/tanx =
   Cos x / sin x
Creating Cotangent Graphs
F(x) = cot x
1. Domain       1.All reals except n(π) where n
is an integer.
2. Range        2.All reals
3. Continuity   3.Continuous on its domain
4. Inc & dec    4.Decreasing on each interval in
its domain.
F(x) = cot x
5. Symmetry     5. Odd
6. Boundedness 6. Unbounded
7. Max/min      7. No max/min
8. Asymptotes   8. No horizontal asymptotes—
VA at nπ where n is an integer.
Cotangent
   Cot x =
   1/tan x
   Since the tangent y-values are…
   0 1 undefined -1 0
   Cotangent y-values are
   Undefined 1 0 -1 undefined
acot(b(x-h)) +k
   a yields a vertical stretch(>1) or
shrink(<1, >0).
   H causes a horizontal translation
   K causes a vertical translation
   B affects the period. Stretch(<1, >0) or
shrink horizontally(>1). Period is π/|b|
Steps for Graphing
Tangent/Cotangent
   Find the period. π/|b|
   Divide the period by 4 and call this x
   Get tick marks for the x-axis and plot
them.
• First one is zero
• Next one is x, then 2x, then 3x, etc.
• 5 points will be one period.
• 4 more points will be another period.
Steps for Graphing
Tangent/Cotangent
   Get y values for each x.
• Tangent y-values are…
• 0 1 undefined -1 0
• Cotangent y-values are…
• Undefined 1 0 -1 undefined
• Take the correct y values and multiply them
by a. These are the y-values to plot for each
x.
   Tangent/Cotangent look like x3.
Secant
   Secant x =
   1/(cos x)=
Creating Secant Graphs
F(x) = sec x
1. Domain
1.All reals except kπ /2 where k
is an odd integer.
2. Range
2.(-∞,-1] U [1,∞)
3. Continuity
3.Continuous on its domain
4. Inc & dec
4. Decreasing on the left half of the
interval containing zero and
increasing on the right half, and
then switches in the next interval.
F(x) = sec x
5. Symmetry     5. even
6. Boundedness 6. Unbounded
7. Max/min      7. Max here corresponds to min of
cos and vice versa.
8. Asymptotes
8. No horizontal asymptotes—
VA-- kπ/2 where k is an odd integer.
asec(b(x-h)) +k
   a yields a vertical stretch(>1) or
shrink(<1, >0).
   H causes a horizontal translation
   K causes a vertical translation
   B affects the period. Stretch(<1, >0) or
shrink horizontally(>1). Period is 2π/|b|
Cosecant
   Cosecant x =
   1/(sin x)=
   Hypotenuse/opposite =
Creating Cosecant Graphs
F(x) = csc x
1. Domain
1. All reals except n π
2. (-∞,-1] U [1,∞)
2. Range
3. Continuous everywhere except at
3. Continuity      n π.
4. Inc & dec    4. Decreasing on the left half of the
“first” interval and increasing on
the right half, and then switches in
the next interval.
F(x) = csc x
5. Symmetry     5. odd
6. Boundedness 6. Unbounded
7. Max/min      7. Max here corresponds to min of
sin and vice versa.
8. Asymptotes
8. No horizontal asymptotes—
VA-- nπ where n is an integer.
acsc(b(x-h)) +k
   a yields a vertical stretch(>1) or
shrink(<1, >0).
   H causes a horizontal translation
   K causes a vertical translation
   B affects the period. Stretch(<1, >0) or
shrink horizontally(>1). Period is 2π/|b|
Steps for Graphing
Secant/Cosecant
   Find the period. 2π/|b|
   Divide the period by 4 and call this x
   Get tick marks for the x-axis and plot
them.
• First one is zero
• Next one is x, then 2x, then 3x, etc.
• 5 points will be one period.
• 4 more points will be another period.
Steps for Graphing
Secant/Cosecant
   Get y values for each x.
•   Secant y-values are 1/(sine y values)
•   Undefined 1 undefined -1 undefined
•   Cosecant y-values are 1/(cosine y values)
•   1 Undefined -1 undefined 1
•   Take the correct y values and multiply them by a.
These are the y-values to plot for each x.
   Secant/Cosecant look like x2.

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