Slide 1-1 The Six Trigonometric Functions Chapter 1.

Slide 1-1

The Six Trigonometric Functions

Chapter 1

Slide 1-2

1.1 Angles, Degree, and Special Triangles

Slide 1-3

1.1 Angles and Degree Measure

Slide 1-4

1.1 Angles and Degree Measure

Angles and arecoterminal if andonly if there is aninteger k such thatm m k360º .

Slide 1-5

1.2 Radian Measure, Arc Length, and Area

Slide 1-6

1.2 Radian Measure, Arc Length, and Area

The radian measure of the angle in standard position is the directed length of the intercepted arc on the unit circle.

Slide 1-7

1.2 Radian Measure, Arc Length, and Area

Slide 1-8

1.2 Radian Measure, Arc Length, and Area

Conversion from degrees to radians or radians to degrees is based on

180 degrees = π radians.

Slide 1-9

1.2 Radian Measure, Arc Length, and Area

Slide 1-10

1.2 Radian Measure, Arc Length, and Area

The length s of an arc intercepted bya central angle of radians on a circleof radius r is given by s r.

Slide 1-11

1.2 Radian Measure, Arc Length, and Area

The area A of a sector with acentral angle of (in radians) in

a circle of radius r is given by Ar2

2.

Slide 1-12

1.3 Angular and Linear Velocity

If a point is in motion on a cirlethrough an angle of radians intime t, then its angular velocity

is given by t

.

Slide 1-13

1.3 Angular and Linear Velocity

If a point is in motion on a circleof radius r through an angle of radians in time t, then its

linear velocity v is given by vst

,

where s is the arc length determinedby s r.

Slide 1-14

1.3 Angular and Linear Velocity

If v is the linear velocity of a pointon a circle of radius r, and is itsangular velociy, then v r .

Slide 1-15

1.4 The Trigonometric Functions

If x, y is any point other thanthe origin on the terminal sideof an angle in standard

position and r x2 y2 , then

sin yr

, cos xr

, tan yx

,

csc ry

, sec rx

, cot xy

,

provided that no denominator is zero.

Slide 1-16

1.4 The Trigonometric Functions

The Reciprocal Identities

csc 1

sin, sec

1cos

, cot 1

tan

Slide 1-17

1.4 The Trigonometric Functions

Table 1.1

Slide 1-18

1.4 The Trigonometric Functions

Slide 1-19

1.4 The Trigonometric Functions

Slide 1-20

1.4 The Trigonometric Functions

Table 1.2

Slide 1-21

1.5 Right Triangle Trigonometry

sin 1 x provided sin xand 90º 90º

cos 1 x provided cos xand 0º 180º

tan 1 x provided tan xand 90º 90º

Slide 1-22

1.5 Right Triangle Trigonometry

Slide 1-23

1.5 Right Triangle Trigonometry

If is an acute angle of a right triangle, then

sin opphyp

, cos adjhyp

, tan oppadj

,

csc hypopp

, sec hypadj

, cot adjopp

.

Slide 1-24

1.5 Right Triangle Trigonometry

Slide 1-25

1.6 The Fundamental Identity and Reference Angles

If is any angle or real number, then

sin2 cos2 1.

Slide 1-26

1.6 The Fundamental Identity and Reference Angles

If is a nonquadrantal angle instandard position, then the referenceangle for is the positive acute angle (read "theta prime") formed by the

terminal side of and the positive ornegative x-axis.

Slide 1-27

1.6 The Fundamental Identity and Reference Angles

Slide 1-28

1.6 The Fundamental Identity and Reference Angles

Slide 1-29

1.6 The Fundamental Identity and Reference Angles

Slide 1-30

1.6 The Fundamental Identity and Reference Angles

For an angle in standard position thatis not a quadrantal anglesin sin , cos cos ,tan tan , cot cot ,csc csc , sec sec where is the reference angle for and the sign is determined by thequadrant in which lies.

Slide 1-31

1.2 The Cartesian Coordinate System

Slide 1-32

P.1 The Cartesian Coordinate System

Slide 1-33

P.1 The Cartesian Coordinate System

Pythagorean TheoremIn a right triangle the sum of thesquares of the legs is equal to the

square of the hypotenuse a2 b2 c2 .

Slide 1-34

P.1 The Cartesian Coordinate System

The Distance Formula

The distance d between the points x1, y1 and x2 , y2 is given by the formula

d x2 x1 2 y2 y1 2 .

Slide 1-35

P.1 The Cartesian Coordinate System

The Midpoint FormulaThe midpoint of the line segment with

endpoints x1, y1 and x2 , y2 is x1 x2

2,y1 y2

2

.