6.3 trigonometric functions (part 1)

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• Length of arc• Angles and measurements (radian

and degree)

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2 2

1 x y

1,0

2

3

2

6

3

4

Denote the unit circle as Uand the length of arc as t

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2 2

1 x y

1,0

3

2

2

6

3

4

Denote the unit circle as Uand the length of arc as t

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Find

12t

1,0

Initial point

Terminal point

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Find 2t

1,0

Initial point

Terminal point

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5

Find

8t

1,0Initial point

Terminal point

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Find 3t

1,0Initial point

Terminal point

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11Find

3t

1,0Initial point

Terminal point

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Angles and Measurements

Definition:

In geometry, angle is thoughtof as union of two rays calledthe sides, having a commonendpoint called the vertex.

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1,0

Initial side

Terminal side

Denote the angle as θ (or any Greek letters)

t

angle θ

Length of arc t

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1,0

Initial side Terminal side

t

angle θ Length of arc t

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Angles and Measurements

Definition:

If the terminal side lies onan axis, the angle is said tobe quadrantal .

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1,0

Initial side

Terminal side

t

quadrantal angle θ

Length of arc t

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Angles and Measurements

Definition:

The measurement of the anglefor which t=1 is called aradian .

It is written as Rm t

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1,0

1 radian

1t

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1,0

2t

2 Rm

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1,0

3.25t 3.25 Rm

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Angles and Measurements

Note that one completerevolution of the terminal sidefrom the initial side in thecounterclockwise direction is2 π .

More than one complete

revolution generates an angleof radian measure greater than2 π .

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1,0

1

4

9

4t

9

4

7

4

Angles having the same terminal side are called coterminal .

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1,0

23

23

t

Find the radian measure of the smallest positive angle that iscoterminal with the angle having the given radian measure.

2 423 3

Rm

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1,0

11

4

114

t

Find the radian measure of the smallest positive angle that iscoterminal with the angle having the given radian measure.

11 324 4

Rm

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1,0

0.54

0.54t

Find the radian measure of the smallest positive angle that iscoterminal with the angle having the given radian measure.

6.28 0.54 5.74 Rm

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Angles and Measurements

Definition:

Another unit of anglemeasurement is the degree .

11 where C is the circumference of a circle

360C

180 radian 1 radian

180

1801 radian

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1,0

1 radian

1t 180

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1 radian180

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Angles and Measurements

Corresponding degree and radianmeasures for certain angles.

Degreemeasure

30 45 60 90 120 135 150 180 270 360

Radianmeasure 16

14

13

12

23

34

56

32

2

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Angles and Measurements

We use the notation toindicate degree measure ofangle θ . It then follows that:

m

180 Rm m

180

Rm m

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Angles and Measurements

Example:

Find the degree measure to thenearest hundredths of a degreefor the angle having the givenradian measure (let π =3.1416).

57

Rm

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Angles and Measurements

Solution: Thus, the anglemeasure of

is

5

7

Rm

180

180 5

7

9007

128.57

R

m m

128.57

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Angles and Measurements

Example:

Find the degree measure to thenearest hundredths of a degreefor the angle having the givenradian measure (let π =3.1416).

0.3826 Rm

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Angles and Measurements

Solution: Thus, the anglemeasure of

is 0.3826 Rm

180

1800.3826

21.92

Rm m

21.92

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Angles and Measurements

We may transform degree anglemeasure which has decimal usingminutes and seconds. That is,

11'

60

1 11'' '

60 3600

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Angles and Measurements

Example:

14 4626 14 '46" 26 60 3600

26 0.233 0.013

26.25

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Assignment (1 whole)

Answer Exercises 6.1 #s 1,3,5,7,15, 17, 19, 21, 23, 25 (page298-299).