5.6 graphs of other trig functions p. 602-603 1-12 all, 55-58 all review table 5.6 on pg 601
TRANSCRIPT
5.6Graphs of Other Trig Functions
p. 602-603 1-12 all, 55-58 allReview Table 5.6 on pg 601
Analysis of the Tangent Function
3 2,3 2 by 4,4
tanf x xDomain: All reals except odd
multiples of 2Range: , Continuous on its domain
Increasing on each interval inits domain
Symmetry: Origin (odd function) Unbounded
No Local Extrema
H.A.: None
V.A.:2
kx
for all odd integers k
End Behavior:
lim tanx
x
and lim tanx
x
do not exist (DNE)
Analysis of the Tangent Function
3 2,3 2 by 4,4
tanf x x
How do we know that these arethe vertical asymptotes?
They are where cos(x) = 0!!!
sin
cos
x
x
How do we know that these arethe zeros?
They are where sin(x) = 0!!!What is the periodof the tangent function???
Period:
Analysis of the Tangent Function
tany a b x h k The constants a, b, h, and k influence the behavior of
in much the same way that they do for the sinusoids…
• The constant a yields a vertical stretch or shrink.
• The constant b affects the period.
• The constant h causes a horizontal translation
• The constant k causes a vertical translation
Note: Unlike with sinusoids, here we do not use theterms amplitude and phase shift…
Analysis of the Cotangent Function
2 ,2 by 4,4
cotf x x
The graph of this function willhave asymptotes at the zerosof the sine function and zerosat the zeros of the cosine function.
Vertical Asymptotes: , 2 , ,0, , 2 ,x
cos
sin
x
x
Zeros:3 3
, , , ,2 2 2 2
x
Guided PracticeDescribe the graph of the given function in terms of a basictrigonometric function. Locate the vertical asymptotes andgraph four periods of the function.
tan 2y xStart with the basic tangent function, horizontally shrink by afactor of 1/2, and reflect across the x-axis.
Since the basic tangent function has vertical asymptotes at allodd multiples of , the shrink factor causes these to moveto all odd multiples of .
24
Normally, the period is , but our new period is . Thus,we only need a window of horizontal length to see fourperiods of the graph…
22
Guided PracticeDescribe the graph of the given function in terms of a basictrigonometric function. Locate the vertical asymptotes andgraph four periods of the function.
tan 2y x
, by 4,4
Guided PracticeDescribe the graph of the given function in terms of a basictrigonometric function. Locate the vertical asymptotes andgraph two periods of the function.
3cot 2 1f x x Start with the basic cotangent function, horizontally stretch bya factor of 2, vertically stretch by a factor of 3, and verticallytranslate up 1 unit.
The horizontal stretch makes the period of the function .2The vertical asymptotes are at even multiples of .
Guided PracticeDescribe the graph of the given function in terms of a basictrigonometric function. Locate the vertical asymptotes andgraph two periods of the function.
3cot 2 1f x x
2 ,2 by 10,10
How would you graph this withyour calculator?
3 tan 2 1y x OR
13 tan 2 1y x
The graph of the secant function secy x 1
cos x
The graph has asymptotes at the zeros of thecosine function.
Wherever cos(x) = 1, its reciprocal sec(x) is also 1.
The period of the secant function is , the sameas the cosine function.
2
A local maximum of y = cos(x) corresponds to alocal minimum of y = sec(x), and vice versa.
The graph of the secant function secy x 1
cos x
22
1
1
The graph of the cosecant function cscy x 1
sin x
The graph has asymptotes at the zeros of the sinefunction.
Wherever sin(x) = 1, its reciprocal csc(x) is also 1.
The period of the cosecant function is , thesame as the sine function.
2
A local maximum of y = sin(x) corresponds to alocal minimum of y = csc(x), and vice versa.
The graph of the cosecant function cscy x 1
sin x
22
1
1
Summary: Basic Trigonometric Functions
Function Period Domain Range
sin x 2 , 1,1
cos x 2 , 1,1
tan x 2x n , cot x x n ,
sec x 2 2x n , 1 1,
csc x 2 x n , 1 1,
Summary: Basic Trigonometric Functions
Function Asymptotes Zeros Even/Odd
sin x ncos x 2 n
tan x 2x n ncot x x n 2 n sec x
csc x
None Odd
None Even
Odd
Odd
2x n
x n
None
None
Even
Odd
Guided Practice
sec 2x 3
2x
Solve for x in the given interval No calculator!!!
Third Quadrant
Let’s construct a reference triangle:sec 2
rx
x
2, 1r x –1
2240x
60 4
3
240
Convert to radians:
Guided Practice
csc 1.5x 3
2x
Use a calculator to solve for x in the given interval.
Third Quadrant
The reference triangle:
1.51
x
1sin
1.5x
csc 1.5r
xy
1.5, 1r y
3.871Does this answer make sense with our graph?
1sin 2 3x
Guided Practice
tan 0.3x 0 2x Use a calculator to solve for x in the given interval.
Possible reference triangles:
0.30.3, 1y x
tan 0.3y
xx
1tan 0.3x 0.291
1tan 0.3x or
3.433
-0.3
-1
1
xx
Whiteboard Problem
sec 2x 3
2x
Solve for x in the given interval No calculator!!!
5
4x
Whiteboard Problem
cot 1x 2
x
Solve for x in the given interval No calculator!!!
3
4x