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Chapter Chapter 4.3 A4.3 ACircular FunctionsCircular Functions

Sine FunctionSine Function

Let be the wrapping function. Then if

sine functio

, ,

the is defn ined assin .

PP t x y

t y

Cosine FunctionCosine Function

Let be the wrapping function. Then if,

cosine functi

,

the is defined as.

oncos

PP t x y

t x

Example 4.3.1Example 4.3.1

Evaluate the following.1. sin 2 0 cos 2 1

2 1,0

2 3 2 12. sin cos

3 2 3 2

2 1 3,

3 2 2

P

P

7 3 7 13. cos sin6 2 6 2

7 3 1,6 2 2

P

Tangent and Cotangent FunctionsTangent and Cotangent Functions

Let be the wrapping function. Then if, ,

then

tan , 0

cot , 0

PP t x y

yt xxxt yy

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Secant and Cosecant FunctionsSecant and Cosecant Functions

Let be the wrapping function. Then if, ,

then1sec , 0

1csc , 0

PP t x y

t xx

t yy

,

sin cos

tan , 0 cot , 0

1 1sec , 0 csc , 0

P t x y

t y t xy xt x t yx y

t x t yx y

Example 4.3.2Example 4.3.2Find the 6 circular function values when .

331. sin

3 212. cos

3 2

323. tan 313 2

1 1 324. cot3 33 3

2

t

1 3,

3 2 2P

Example 4.3.2Example 4.3.2

Find the 6 circular function values when .3

t

1 3,

3 2 2P

5. sec 2

3

2 2 36. csc3 33

Example 4.3.3Example 4.3.3

Find the 6 circular function values when .1. sin 0

2. cos 10

3. tan 01

4. cot is undefined15. sec 11

6. csc is undefined

t

1,0P

Challenge!Challenge!

Evaluate the following as fast as you can.2 1 41. cos 5. sec 23 2 3

2. tan 1 6. csc 0 is undefined4

11 1 5 23. sin 7. sin6 2 4 2

34. cot 3 8. tan6 6 3

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-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-2

-1

1

2

t

f (t)

x

y

Domain, Range, and GraphsDomain, Range, and Graphs

1. sin

1,1

f t t

Dom f

Rng f

,

sin

P t x y

t y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-2

-1

1

2

t

f (t)

x

y

Domain, Range, and GraphsDomain, Range, and Graphs

2. cos

1,1

f t t

Dom f

Rng f

,

cos

P t x y

t x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3

-2

-1

1

2

3

t

f (t)

Domain, Range, and GraphsDomain, Range, and Graphs

3. tan

, is an odd integer2

f t t

kDom f t t k

Rng f

, tan yP t x y t x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3

-2

-1

1

2

3

t

f (t)

Domain, Range, and GraphsDomain, Range, and Graphs

4. cot

, is an integer

f t t

Dom f t t k k

Rng f

, cot xP t x y t y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3

-2

-1

1

2

3

t

f (t)

Domain, Range, and GraphsDomain, Range, and Graphs

5. sec

, is an odd integer2

, 1 1,

f t t

kDom f t t k

Rng f

1, secP t x y t x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3

-2

-1

1

2

3

t

f (t)

Domain, Range, and GraphsDomain, Range, and Graphs

6. csc

, is an integer

, 1 1,

f t t

Dom f t t k k

Rng f

1, cscP t x y t y

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Periodic FunctionsPeriodic Functions

are functions whose valuesrepeat after a fixed interval.

The fixed interval is c

Periodic

alled the

functions

pe riod.-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

-2

-1

1

2

x

f (t)

Periodic FunctionsPeriodic Functions

The six circular functions are periodic.

period

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

-2

-1

1

2

x

f (t)

CycleCycle

A is simply the portion of a graph of afunction over an interval of length equalto the per

cycle

iod.

1 cycle 1 cycle 1 cycle

AmplitudeAmplitude

The of a graph of the sine or cosine function refers to one-half the absolute difference between the highest and the lowest function

amplitud

valu

e

es.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-2

-1

1

2

t

f (t)

AmplitudeAmplitude

1 1amplitude 1 1 2 12 2

Sine and Cosine FunctionsSine and Cosine Functions

Form: sin

cosDomain:

Range: ,

2Amplitude: Period:

f x a b x h k

g x a b x h k

k a k a

ab