PRECALCULUSLESSON

Check: p. 326 [Evens] # 12-18, 32-36, 50-54

4.6

WARM-UP

1. ( ) 4sin2. ( ) sin(2 ) 2

3. ( ) 3cos2

f x xg x x

h x x

Graph 2 full periods.

ANSWERS

On other sheet

OBJECTIVE:

Graph sine cosine and tangent functions, and their reciprocals.

Graphs of Trigonometric Functions

4.6 Digital Lesson

Tangent, cotangent, secant, and cosecant

y

x

23

23

2

2

Graph of the Tangent Function

2. range: (–, +) 3. period: 4. vertical asymptotes:

kkx 2

1. domain : all real x kkx

2

Properties of y = tan x

period:

To graph y = tan x, use the identity .xxx

cossintan

At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.

2. Find consecutive vertical asymptotes by solving for x:

4. Sketch one branch and repeat.

Example: Find the period and asymptotes and sketch the graph of xy 2tan

31

22 ,

22

xx

4 ,

4

xxVertical asymptotes:

)2

,0( 3. Plot several points in

1. Period of y = tan x is.

2 . is 2tan of Period xy

x

xy 2tan31

8

31

0

08

31

83

31

y

x2

83

4

x4

x

31,

8

31,

8

31,

83

Graph of the Cotangent Function

2. range: (–, +) 3. period: 4. vertical asymptotes:

kkx

1. domain : all real x kkx

Properties of y = cot x y

x

2

2

23

23

2

xy cot

0xvertical asymptotes xx 2x

To graph y = cot x, use the identity .xxx

sincoscot

At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.

Example 1: Graph a) tan2xy

Consecutive Asymptotes 2 2

bx c

Period: difference between consecutive asymptotes

Add or subtract the Period to Consecutive asymptotes to

complete 2 periods.

Graph 2 full periods (cycles)• Asymptotes and intercepts.

Example 1:Graph b)

0 bx c

Period: difference between consecutive asymptotes

Add or subtract the Period to Consecutive asymptotes to

complete 2 periods.

cot2

y x

Consecutive Asymptotes

Graph 2 full periods (cycles)• Asymptotes and intercepts.

23

y

x

2

2

2 32

5

4

4

xy cos

Graph of the Secant Function

2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes:

kkx 2

1. domain : all real x)(

2 kkx

cos

1secx

x The graph y = sec x, use the identity .

Properties of y = sec x

xy sec

At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

23

x

2

2

22

5

y4

4

Graph of the Cosecant Function

2. range: (–,–1] [1, +) 3. period:

where sine is zero.

4. vertical asymptotes: kkx

1. domain : all real x kkx

sin

1cscx

x To graph y = csc x, use the identity .

Properties of y = csc x xy csc

xy sin

At values of x for which sin x = 0, the cosecant functionis undefined and its graph has vertical asymptotes.

Example 2 graph a)

:amplitude a

endpoints for one period: 0 bx-c 2

2:periodb

x

y

:4

periodInterval

2csc4

y x

Table for 2sin4

y x

Graph 2 full periods (cycles)x

y

Example 2 graph b)

:amplitude a

endpoints for one period: 0 bx-c 2

2:periodb

x

y

:4

periodInterval

3sec2y x

Table for 3cos2y x

Graph 2 full periods (cycles)x

y

Classwork: p. 337 # 17, 33, 35

Homework(4.6)(4.6) p. 337 # 16, 18, 22, 34(4.5) p. 327 # 40, 42, 56, 60

Assignments Unit Circle QuizTomorrow

No Calculator

Find the amplitude & period for each.1. f(x) = 2cos 4x2. G(x) = -5sin 2x3. H(x) = tan x4. P(x) = 3 cot 2x

CLOSURE