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