-p

2

p

2p 3p

22p 5p

2

-2

-1

1

2

x

y

-p

2

p

2p 3p

22p 5p

2

-2

-1

1

2

x

y

-p

2

p

2p 3p

22p 5p

2

-2

-1

1

2

x

y

𝒚=𝐬𝐢𝐧 𝒙𝒚=𝐜𝐨𝐬 𝒙

𝒚=𝐬𝐢𝐧 𝒙+𝐜𝐨𝐬𝒙𝒚=𝐬𝐢𝐧 𝒙𝐜𝐨𝐬 𝒙𝒚=𝐬𝐢𝐧 𝒙𝐜𝐨𝐬 𝒙

13.6The Tangent Function

Today’s Objective:

I can find exact values of the tangent function.

𝒚=𝐭𝐚𝐧 𝒙

x

y

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

1y

x

tan Opp.Adj.

yx,

SOH CAH TOA

xy

tancos

sin

y sinx cos

tan = Sloperunrise

Tangent of θ or

6

tanp

x

y

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

2

1

2

3

3

3

6

7tan

p3

3

sincos

30o210o

- 30o

6

5tan

p3

3-

6

11tan

p3

3-

2

1

3

2

3

1 3

3

3

tanp

x

y

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

2

3

2

1

3

3

4tan

p

sincos

60o

3

3

2tan

p3-

3

5tan

p3-

2

3

1

2

- 60o

4

tanp

x

y

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

2

2

2

2

1

4

5tan

p

sin

cos

45o

- 45o

1

4

3tan

p1-

4

7tan

p1-

2

tanp

x

y

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

10

undefined

2

3tan

p

sincos

0

ptan0tan

0

1

0

undefined

p. 871: 1-4, 9-15 odd

Day 213.6

The Tangent Function

Today’s Objective:

I can graph the tangent function.

𝒚=𝐭𝐚𝐧 𝒙

-p -3p

4

-p

2

-p

4

p

4

p

2

3p

4p

-3

-2

-1

1

2

3

x

y

x

y

Domain:

Range:

Period:

x

y

(1, 0)

(0, 1)

(0, -1)

𝐭𝐚𝐧𝝅𝟔

=¿¿

𝐭𝐚𝐧 𝝅𝟒

=¿¿

𝐭𝐚𝐧𝝅𝟑

=¿¿

𝐭𝐚𝐧−𝝅𝟔

=¿¿

𝐭𝐚𝐧−𝝅𝟒

=¿¿

𝐭𝐚𝐧−𝝅𝟑

=¿¿

√𝟑𝟏√𝟑𝟑

− √𝟑𝟑−𝟏

−√𝟑

𝐭𝐚𝐧𝟎=¿¿𝟎

All real #s except odd multiples of

All Real #sπ

x

y

x

y

x

y

-p -3p

4

-p

2

-p

4

p

4

p

2

3p

4p

-3

-2

-1

1

2

3

x

y

-p -3p

4

-p

2

-p

4

p

4

p

2

3p

4p

-3

-2

-1

1

2

3

x

y

Graphing Tangent Functions

𝑦=𝒂 tan𝒃𝑥 Period = Asymptotes = ; then end of each period.a = vertical stretch

Sketch the graph of: 𝑦=tan𝟐𝑥Graphing:1. Determine period & graph asymptotes2. Graph middle points3. Graph halfway points:

between asymptotes & middle pointVertical distance a and – a

Period: Asymptote: 𝜋2(2)

x

y

x

y

¿𝜋4

𝜋2

x

y

x

y

x

y

-2p -3p

2-p -p

2

p

2p 3p

22p

-3

-2

-1

1

2

3

x

y

-2p -3p

2-p -p

2

p

2p 3p

22p

-3

-2

-1

1

2

3

x

ySketch the graph of:

Period: Asymptote 𝜋2(0.5)¿𝜋2𝜋

from -2π to 2π

x

y

x

y

x

y

x

y

Graphing Tangent Functions

𝑦=𝒂 tan𝒃𝑥 Period = Asymptotes = ; then end of each period.a = vertical stretch

Graphing:1. Determine period & graph asymptotes2. Graph middle points3. Graph halfway points:

between asymptotes & middle pointVertical distance a and – a

𝑦=tan12𝑥

Pg. 872 #9-25odds, 30-32,44-46

W.S. Trigonometric Functions

-3p -2p -p p 2p 3p-3

-2

-1

1

2

3

x

y