lat03 01

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(2)

Chapter 1

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1.1

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Basic Terms

 Two distinct points determine a line called

line AB.

 Line segment AB—a portion of the line

between A and B, including points A and B.

 Ray AB—portion of line AB that starts at A and

continues through B, and on past B.

A B

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Basic Terms continued

 Angle-formed by rotating

a ray around its endpoint.

 The ray in its initial

position is called the

initial side of the angle.

 The ray in its location

after the rotation is the

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Basic Terms continued

 Positive angle: The

rotation of the terminal side of an angle

counterclockwise.

 Negative angle: The

(7)

Types of Angles

 The most common unit for measuring angles is

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Example: Complementary Angles

 Find the measure of each angle.

 Since the two angles form a right

angle, they are complementary angles. Thus,

k 16

k +20

20 16

2

9

4

8

4

2 6

0

90

3

k k

k

k k

   

  

(9)

Example: Supplementary Angles

 Find the measure of each angle.

 Since the two angles form a straight

angle, they are supplementary angles. Thus,

6x + 7 3x + 2

6 7 3 2

9 9

180

180 9 171

19

x x

x

x x

   

  

(10)

Degree, Minutes, Seconds

 One minute is 1/60 of a degree.

 One second is 1/60 of a minute.

or

1

1' 60' 1

60

   

1 1

1" 60" 1' 60 3600 or

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Example: Calculations

 Perform the calculation.

 Since 86 = 60 + 26, the

sum is written

 Perform the calculation.

 Write

27 34' 26 52'  

72





15 18'



27 34' 26 52' 53 86'     53 1 26' 54 26'    

71 60

15 18'

56 42'



  



(12)

Example: Conversions

 Convert

74 12' 18"

  Convert 36.624

18 18"

3600

74 74

74 .2 .00

12 12' 6 5 7 0 4.205             

34.624 34 .624

34 .624(60') 34 37.44' 34 37' .44'

34 37' .44(60") 34 37' 26.4" 34 37' 26.4"

(13)

Standard Position

 An angle is in standard position if its vertex is

at the origin and its initial side is along the positive x-axis.

 Angles in standard position having their terminal

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Coterminal Angles

 A complete rotation of a ray results in an angle

measuring 360. By continuing the rotation, angles of measure larger than 360 can be

(15)

Example: Coterminal Angles

 Find the angles of smallest possible positive

measure coterminal with each angle.

 a) 1115 b) 187

 Add or subtract 360 as may times as needed to

obtain an angle with measure greater than 0 but less than 360.

 a) b) 187 + 360 =

173

(16)

1.2

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Vertical Angles

 Vertical Angles have equal measures.

 The pair of angles NMP and RMQ are vertical

angles.

M Q

R

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Parallel Lines

 Parallel lines are lines that lie in the same plane

and do not intersect.

 When a line q intersects two parallel lines, q, is

called a transversal.

m

n

parallel lines

q

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Angles and Relationships

m

n q

Angle measures are equal. 2 & 6, 1 & 5,

3 & 7, 4 & 8 Corresponding angles

Angle measures add to 180. 4 and 6

3 and 5 Interior angles on the same

side of the transversal

Angle measures are equal. 1 and 8

2 and 7 Alternate exterior angles

Angles measures are equal. 4 and 5

3 and 6 Alternate interior angles

Rule Angles

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Example: Finding Angle Measures

 Find the measure of each

marked angle, given that

lines m and n are parallel.

 The marked angles are

alternate exterior angles, which are equal.



 One angle has measure

6x + 4 = 6(21) + 4 = 130

 and the other has measure

10x  80 = 10(21)  80 =

130 m

n

(10x  80)

(6x + 4)

6 4 10 80

84 4 21

  

 

x x

(21)

Angle Sum of a Triangle

 The sum of the measures of the angles of any

(22)

Example: Applying the Angle Sum

 The measures of two of

the angles of a triangle

are 52 and 65. Find the

measure of the third

angle, x.

 Solution

 The third angle of the

triangle measures 63.

52

65

x

52 65 11

180

7 180

63

  

  

  

 

 x

x

(23)

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(25)

Conditions for Similar Triangles

 Corresponding angles must have the same

measure.

 Corresponding sides must be proportional.

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Example: Finding Angle Measures

 Triangles ABC and DEF

are similar. Find the

measures of angles D

and E.

 Since the triangles are

similar, corresponding angles have the same measure.

 Angle D corresponds to

angle A which = 35

 Angle E corresponds to

angle B which = 33

A

C B

F E

D

35

112 33

(27)

Example: Finding Side Lengths

 Triangles ABC and DEF

are similar. Find the

lengths of the unknown sides in triangle DEF.

 To find side DE.

 To find side FE.

A

C B

F E

D

35

112 33

(28)

Example: Application

 A lighthouse casts a

shadow 64 m long. At the same time, the shadow cast by a mailbox 3 feet high is 4 m long. Find the height of the lighthouse.

 The two triangles are

similar, so corresponding sides are in proportion.

 The lighthouse is 48 m

high.

3

4 64

4 192

48



 

x

x x

64 4

3

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1.3

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Trigonometric Functions

 Let (x, y) be a point other the origin on the

terminal side of an angle  in standard position. The distance from the point to the origin is

The six trigonometric functions of  are defined as follows.

2 2_.

r  x  y

sin y cos x tan y (x 0)

r r x

      

csc r (y 0) sec r (x 0) cot x (y 0)

y x y

(31)

Example: Finding Function Values

 The terminal side of angle  in standard position

passes through the point (12, 16). Find the values of the six trigonometric functions of angle .

(12, 16)

16

12



2 2 2 ₁₆2

144 256 0 2

1

0 2

40

r  x  y  

(32)

Example: Finding Function Values

continued

 x = 12 y = 16 r = 20

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Example: Finding Function Values

 Find the six trigonometric

function values of the

angle  in standard

position, if the terminal

side of  is defined by

x + 2y = 0, x  0.

 We can use any point on

the terminal side of  to

(34)

Example: Finding Function Values

continued

 Choose x = 2

 The point (2, 1) lies on

the terminal side, and the

corresponding value of r

is

 Use the definitions:

2 0 2 2 2 1 2 0 y y x y y         2 2

2 ( 1) 5.

r    

1 1 5 5 5 5 5 5

2 2 5 2 5 5 5 5 5

1 5 2 sin cos tan csc

sec 5 cot 2

(35)

Example: Function Values Quadrantal

Angles

 Find the values of the six trigonometric functions for an angle

of 270.

 First, we select any point on the terminal side of a 270 angle.

We choose (0, 1). Here x = 0, y = 1 and r = 1.

1 0

sin 270 1 cos 270 0

1 1

tan 270 undefined csc 270 1

0 1

1 0

sec270 undefined cot 270 0

0 1



    



   



  

 

(36)

Undefined Function Values

 If the terminal side of a quadrantal angle lies

along the y-axis, then the tangent and secant functions are undefined.

 If it lies along the x-axis, then the cotangent and

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Commonly Used Function Values

undefined 1 undefined 0 1 0 360 1 undefined 0 undefined 0 1 270 undefined 1 undefined 0 1 0 180 1 undefined 0 undefined 0 1 90 undefined 1 undefined 0 1 0 0

csc 

sec 

cot 

tan 

cos 

sin 

(38)

1.4

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Reciprocal Identities

 _sin 1 _cos 1 _tan 1

csc sec cot

1 1 1

csc sec cot

sin cos tan

  

  

(40)

Example: Find each function value.

 cos  if sec  =

 Since cos  is the

reciprocal of sec 

 sin  if csc 

2 3

1 1 2

cos sec 3      15 3   15 3 1 sin 3 15 3 15 15 15 3 15 15

(41)

Signs of Function Values

 +   +  IV   + +   III +     + II + + + + + + I

csc 

sec 

cot 

tan 

cos 

sin 

(42)

Example: Identify Quadrant

 Identify the quadrant (or quadrants) of any angle

 that satisfies tan  > 0, cot  > 0.

 tan  > 0 in quadrants I and III  cot  > 0 in quadrants I and III

(43)

Ranges of Trigonometric Functions

 For any angle  for which the indicated functions

exist:

 1. 1  sin   1 and 1  cos   1;

 2. tan  and cot  can equal any real number;  3. sec   1 or sec   1 and

csc   1 or csc   1.

(Notice that sec  and csc  are never between

(44)

Identities



Pythagorean



Quotient

2 2

sin

cos

1,

tan

1 sec ,

1 cot

csc







 





sin

tan cos

cos

cot sin

 _



 _





(45)

Example: Other Function Values

 Find sin and cos if tan  = 4/3 and  is in

quadrant III.

 Since is in quadrant III, sin and cos will both

be negative.

(46)

Example: Other Function Values

continued

 We use the identity _tan2  _{1 sec}2

2 2

2

tan 1 sec

1 s 3 ec 16 1 sec 9 25 sec 9 5 sec 3 cos 4 5 3                         2 2 2 2 2 2 4 5

Since sin 1 cos ,