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4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

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Page 1: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

4.5

Graphs of Tangent, Cotangent, Secant, and Cosecant

Page 2: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Quiz: 4-4 2433sin5.0)( xxf

1. Vertical stretch/shrink1. Vertical stretch/shrink2. Horizontal stretch/shrink2. Horizontal stretch/shrink3. Horizontal translation (phase shift)3. Horizontal translation (phase shift)4. Vertical translation 4. Vertical translation

Page 3: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

What you’ll learn about

• The Tangent Function• The Cotangent Function• The Secant Function• The Cosecant Function

… and whyThis will give us functions for the remaining trigonometric ratios.

Page 4: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

UNIT CIRCLE

r = 1

3030ºº

AA

CC

6060ºº

x

y

BB

3030ººAA

CC

BB

16060ºº

1

3

01

00tan2tan

x

ytan

2

2

33

232

130tan

6tan

Page 5: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

UNIT CIRCLE

-60-60ºº

undefined 0/1)90tan( x

ytan

3)60tan(

-30-30ºº

6060ºº3030ºº00ºº 9090ºº

11

-90-90ºº

Vertical asymptoteVertical asymptote

-1-1

33)30tan(

00tan

3330tan

360tan

0190tan

Vertical asymptoteVertical asymptote

Page 6: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

UNIT CIRCLE

x

ytan

00ºº 9090ºº

11

-90-90ºº

-1-1

Pattern repeatsPattern repeats

Page 7: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Your turn:Your turn:1.1. Domain = ?Domain = ?2.2. Range = ?Range = ?3.3. Continuous?Continuous?4.4. Symmetry?Symmetry?5.5. bounded?bounded?6.6. Vertical/horizontal asymptotes?Vertical/horizontal asymptotes?7.7. End behavior?End behavior?

Your turn:

0)(lim xfx

0)(lim xfx

Page 8: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

f(x) = sin x

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

radians

radians

Remember this?Remember this?

The The y-coordinatey-coordinate of the graph is the of the graph is the distance above/belowdistance above/below the x-axis on the unit circle.the x-axis on the unit circle.

The The x-coordinatex-coordinate of the graph is the of the graph is the distance arounddistance around the unit circle. the unit circle.

Same thingSame thing for for cos x.cos x.

Page 9: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

How do we construct the Tangent Function’s graph?

Sin xSin x

Cos xCos x

y = Sin xy = Sin x: : the verticalthe vertical distance above/belowdistance above/below the x-axis (on the unit circle).the x-axis (on the unit circle).

x = cos xx = cos x: : the horizontalthe horizontal distance to the right/left ofdistance to the right/left of the y-axis (on the unit circle).the y-axis (on the unit circle).

This distance is graphed asThis distance is graphed as the y-coordinate on this graph.the y-coordinate on this graph.

This distance is also graphed asThis distance is also graphed as the y-coordinate on this graph.the y-coordinate on this graph.

Page 10: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Construct the Tangent Function using the graph of ‘y’ and ‘x’

)(

)(tan

lengthadjacent

lengthoppositeA

x

yA tan

y = Sin xy = Sin x

x = Cos xx = Cos x

y = Sin xy = Sin x: (on unit circle): (on unit circle)

x = cos xx = cos x: (on unit circle): (on unit circle)

Using the graphs of the left:Using the graphs of the left:

1

00tan

0

190tan

Page 11: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

The Tangent and Cotangent Functions

Tan xTan x

Cot xCot x

tan

1cot

Page 12: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Asymptotes and Zeros of the Tangent Function

x

xx

cos

sintan

b

Rules for Tangent: y = tan (bx – c) + d

a = none

period =

A full period of tan occurs between 2 consecutive asymptotes.

To find 2 consecutive asymptotes: bx – c = and bx – c =

The graph starts low, ends high, ½ way thru period is where it crosses the x-axis.

X-intercepts: sin X-intercepts: sin x x = 0= 0

Asymptotes: cos Asymptotes: cos xx = 0 = 0

2

b

2

Page 13: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

The Tangent Function

x

xx

cos

sintan

2

X-intercepts: sin X-intercepts: sin x x = 0= 0

Asymptotes: cos Asymptotes: cos xx = 0 = 0

2

b

2

AmplitudeAmplitude: doesn’t have one (look at the graph): doesn’t have one (look at the graph)

dcbxxf )tan()(

PeriodPeriod: : A full period of tan(x) occurs between 2 A full period of tan(x) occurs between 2 asymptotes which occur at asymptotes which occur at oddodd multiples of multiples of

Shrink factor causes asymptotes to be odd multiplies of: Shrink factor causes asymptotes to be odd multiplies of: b2

Page 14: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Describe how tan(x) is transformed to graph: -tan(2x)

dcbxxf )tan()(

b

2

1. 1. AmplitudeAmplitude: n/a: n/a

2. 2. PeriodPeriod: :

3. 3. Vertical AsymptotesVertical Asymptotes: odd multiples of: odd multiples of b2

4

4. -1 coefficient: causes reflection across x-axis.4. -1 coefficient: causes reflection across x-axis.

Tan(x)Tan(x) -tan(2x)-tan(2x)

Page 15: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

The Sine and Cosecant Functions

sin

1csc

Sine (x)Sine (x)

Cosecant (x)Cosecant (x)

Page 16: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

The Cosecant Function: y = 2 csc 2x

To graph graphing shifts, rewrite the equation as it’s reciprocal, then find zero’s, max/mins (amp), and period.

dcbxxf

)sin(

1)(

1. Vertical stretch: factor of 21. Vertical stretch: factor of 2

)2sin(

2)(

xxf

sin(x) csc(x) sin(x) csc(x)

2. Vertical asymptotes: even multiples of2. Vertical asymptotes: even multiples of .,2*2,2*02

etc

3. Horizontal shrink factor of 3. Horizontal shrink factor of ½, changes period to ½, changes period to 2csc(2x)2csc(2x)

Page 17: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

The Cosine and Secant Functions

Cos xCos x

cos

1sec

Sec xSec x

Page 18: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Solve Trig function AlgebraicallyFind: ‘x’ such that and sec(x) = -2 Find: ‘x’ such that and sec(x) = -2 2

3 x

2)sec( x

rx

1

22

x

r

--½½

11

23

Converting to unit circle:Converting to unit circle:

r = 1 x = -r = 1 x = -½½

3

3030ºº

1 6060ºº½

2

That looks familiar!That looks familiar!

6060ºº

240240ºº x = 240x = 240ºº 34

Page 19: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Solving a Trig equation graphically

Find the smallest number ‘x’ such that:Find the smallest number ‘x’ such that: xx csc2 1. Graph and 1. Graph and

2xy )csc(xy

2. Find the point of intersection2. Find the point of intersection

068.1x

Page 20: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Basic Trigonometry Functions

Page 21: 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

HOMEWORK

Section 4-5