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

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.

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

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

UNIT CIRCLE

x

ytan

00ºº 9090ºº

11

-90-90ºº

-1-1

Pattern repeatsPattern repeats

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

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.

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.

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

The Tangent and Cotangent Functions

Tan xTan x

Cot xCot x

tan

1cot

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

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

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)

The Sine and Cosecant Functions

sin

1csc

Sine (x)Sine (x)

Cosecant (x)Cosecant (x)

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)

The Cosine and Secant Functions

Cos xCos x

cos

1sec

Sec xSec x

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

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

Basic Trigonometry Functions

HOMEWORK

Section 4-5