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Slide Angles and Degree Measure
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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
.