t2_trig fn of real numbers unit circle

8/9/2019 T2_trig Fn of Real Numbers UNIT Circle

1/23

The Unit Circle

One of the most useful tools in trigonometry is the unit circle.

It is a circle, with radius 1 unit, that is on the x-y coordinate plane.

The angles are measured from the positive

x-axis (standard position) counterclockwise.

The x-axis corresponds to the cosine function, and the y-

axis corresponds to the sine function.

122! yx


2/23

The Unit Circle (contd)

The formulas = rwith r= 1 and = tgives

s = r= 1(t) = t

Thus, tmay be regarded either as

the radian measure of the angle or as

the length of the circular arcAPon U.


3/23


Next consider any nonnegative real numbert.

If we regard the angle of radian measure tas

having been generated by rotating the linesegment OA about O in the counterclockwise

direction, then tis the distance along UthatA

travels before reaching its final position:


4/23



5/23


Ift< 0, then the rotation ofOA is in the

clockwise direction, and the distanceA travels

is |t|. Thus we can associate with each real numbert

a unique pointP(x, y) on U.

We shall callP(x, y) the pointon the unit

circle U that corresponds to t.


6/23

Trigonometric Functions

Our earlier definitions of the trigonometric

functions lead to the following formulas:

We sometimes refer to the trigonometric

functions as the circular functions.


7/23

15tan

4

1cos

4

15sin

)4

15,

4

1(),(

!

!!

!!

!

t

xt

yt

yx

15

1

tan

1cot

15

41csc

41

sec

!!

!!

!!

tt

yt

xt

Example:

Solution:


8/23

Example

LetP(t) denote the point on Uthat

corresponds to tfor 0 t< 2.

IfP(t) = (, ), find

a) P(t+ )

b) P(t)

c) P(t)


9/23

Example (contd)


10/23

Example (contd)


11/23

If there is a smallest such number

p, this smallest value is called the

(fundamental) period off.

Definition:


12/23

E

xample of periodic functions.E

xample of periodic functions.

1. The sine function.y = sin t

@sin ( t+ 2n

T) = sin tfor any integern ,

smallest positive integer , n = 1sin ( t+ 2T ) = sin t

period is 2T

[ same fory = csc t]


13/23

2 T

T /2

T

3 T /2

(1, 0)(1, 0)

(0, 1)

(0, 1)

0

1

P(cos t , sin t )

ab b

a

x

a

b

x

y

1

-1

0 T 2T 3T 4TT2T

Period: 2T

Domain: All real numbers

Range: [1, 1]

Symmetric with respect to the origin

Graph ofy = sin t

y = sin t= b


14/23

E



2. The cosine function.y = cos t

@cos ( t+ 2n

T) = cos tfor any integern ,

smallest positive integer , n = 1cos ( t+ 2T ) = cos t

period is 2T[ same fory = sec t ]


15/23

2 T

T /2

T

3 T /2

(1, 0)(1, 0)

(0, 1)

(0, 1)

0

1

P(cos t , sin t )

ab b

a

x

a

b

x

y

1

-1

0 T 2T 3T 4TT2T

Period: 2T


Range: [1, 1]

Symmetric with respect to the

y axisy = cos t = a

Graph ofy = cos t

5-6-57


16/23

E



3. The tangent function.y = tan t

@tan ( t+ n

T) = tan tfor any integern ,

smallest positive integer , n = 1tan ( t+ T ) = tan t

period is T[ same for y = cot t]


17/23

x

y

T 2 T2 T T 0

1

1

5T2

3 T2

T2

5 T2

3 T2

T2

Period: T


except T /2 + kT,k

an integer

Range: All real numbers

Symmetric with respect to

the origin

Increasing functionbetween asymptotes

Discontinuous at

x= T /2 + kT, kan integer

Graph ofy = tanx

5-6-58


18/23

Cosecant Function


19/23

Secant Function


20/23

Cotangent Function


21/23

Periodic Properties


22/23

Theorem Even-Odd Properties


23/23

Assignment

Exercises 1 page 355

Exercises 5 page 355

Home work

Exercises 5-8, 26, 35, 36 46 page 348 Exercises 59 page 350

t2_trig fn of real numbers unit circle

Documents