College Trigonometry 2 Credit hours through KCKCC or Donnelly

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Chapter 1Trigonometric Functions
Section 1.1 Basic Concepts
Section 1.2 Angles
Section 1.3 Angle Relationships
Section 1.4 Definitions of Trig Functions
Section 1.5 Using the Definitions
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Section 1.1 Basic Concepts

• In this section we will cover
• Labeling Quadrants
• Pythagorean Theorem
• Distance Formula
• Midpoint Formula
• Interval Notation
• Relations
• Functions

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The Coordinate Plane

• Horizontal
• x
• abscissa
• Vertical
• y
• ordinate

Quadrant I (,)
Quadrant II (-,)
Quadrant III (-,-)
Quadrant IV (,-)
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Pythagorean Theorem

• a2 b2 c2

B
hypotenuse c
leg a
A
C
leg b
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Distance Formula

• a (x2 x1)
• b (y2 y1)
• c v a2 b2
• or
• distance v (x2 x1)2 (y2 y1)2

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Midpoint Formula

• The Midpoint Formula The midpoint of a segment
  with endpoints (x1 , y1) and (x2 , y2) has
  coordinates

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Interval Notation

• Set-builder notation
• xxlt5 the set of all x such that x is less than
  5
• Interval notation
• (-8, 5) the set of all x such that x is less than
  5
• (-8, 5 the set x is less than or equal to 5
• the first is an open interval
• the second is a half-opened interval
• 0, 5 is an example of a closed interval

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Relations and Functions

• A relation is a set of points.
• A dependent variable varies based on an
  independent variable.
• For example y 2x y is the dependent variable
• x is the
  independent variable
• A relation is a function if each value of the
  independent variable leads to exactly one value
  of the dependent variable.

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• The values of the dependent variable represent
  the range.
• The values of the independent variable represent
  the domain.
• A relation is a function if a vertical line
  intersects its graph in no more than one point.
  (Vertical Line Test)

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Section 1.2 Angles

• In this section we will cover
• Basic terminology
• Degree measure
• Standard position
• Co terminal Angles

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Basic Terminology

• line - an infinitely-extending one-dimensional
  figure that has no curvature
• segment - the portion of a line between two
  points
• ray - the portion of a line starting with a
  single point and continuing without end
• angle - figure formed through rotating a ray
  around its endpoint

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Basic Terminology (cont)

• initial side - ray position before rotation
• terminal side - ray position after rotation
• vertex - point of rotation
• positive rotation - counterclockwise rotation
• negative rotation - clockwise rotation
• degree - 1/360th of a complete rotation

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Basic Terminology (cont)

• acute angle - angle with a measure between 0 and
  90
• right angle - angle with a measure of 90
• obtuse angle - angle with a measure between 90
  and 180
• straight angle - angle with a measure of 180
• complementary - sum of 90
• supplementary - sum of 180

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Basic Terminology (cont)

• minute - , 1/60th of a degree
• second , 1/60th of a minute, 1/3600th of a
  degree
• standard position - an angle with a vertex at the
  origin and initial side on the positive abscissa
• quadrantal angles - angles in standard position
  whose terminal side lies on an axis
• co terminal angles - angles having the same
  initial and terminal sides but different angle
  measures

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Section 1.3 Angle Relationships

• In this section we will cover
• Geometric Properties
• Vertical angles
• Parallel lines cut by a transversal
• Corresponding angles
• Same side interior and exterior angles
• Applying triangle properties
• Angle sum
• Similar triangles

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Geometric Properties

• Vertical angles are formed when two lines
  intersect. They are congruent which means they
  have equal measures.
• When parallel lines are cut by a third line,
  called a transversal, the result is to sets of
  congruent angles.

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Geometric Properties (cont

• So here angles 1, 4, 5, and 8 are congruent and
  angles 2, 3, 6, and 7 are congruent.
• Corresponding pairs are / 1 / 5, / 2 / 6,
• / 3 / 7, and / 4 / 8.

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Triangle Properties

• The sum of the interior angles of a triangle
  equal 180.
• Acute 3 acute angles
• Right 2 acute and one right angle
• Obtuse 1 obtuse and two acute angles
• Equilateral all sides (and angles) equal
• Isosceles two equal sides (and angles)
• Scalene no equal sides (or angles)

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Triangle Properties (cont)

• Corresponding parts of congruent triangles are
  congruent.
• Corresponding angles of similar triangles are
  congruent.
• Corresponding sides of similar triangles are in
  proportion.

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Section 1.4 Definitions of Trigonometric Functions

• In this section we will cover
• Trigonometric functions
• Sine
• Cosine
• Tangent
• Quadrantal angles

• Cosecant
• Secant
• Cotangent

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Trigonometric Functions

• Sine opposite /hypotenuse y/r
• Cosine adjacent/hypotenuse x/r
• Tangent opposite/adjacent y/x
• Cosecant hypotenuse/opposite r/y
• Secant hypotenuse/adjacent r/x
• Cotangent adjacent/opposite x/r

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Special TrianglesSpecial Trig Values
30à 45à 60à 90à
sin 1/2 ñ2/2 ñ3/2 1
cos ñ3/2 ñ2/2 1/2 0
tan ñ3/3 1 ñ3 Und
csc 2 ñ2 2ñ3 3 1
sec 2ñ3 3 ñ2 2 Und
cot ñ3 1 ñ3/3 0
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Trigonometric Functions Values for Quadrant Angles
0à 90à 180à 270à
sin 0 1 0 -1
cos 1 0 -1 0
tan 0 Undefined 0 Undefined
csc Undefined 1 Undefined -1
sec 1 Undefined -1 Undefined
cot Undefined 0 Undefined 0
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Section 1.5 Using the Definitions of
Trigonometric Functions

• In this section we will cover
• The reciprocal identities
• Signs and ranges of function values
• The Pythagorean identities
• The quotient identities

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The Reciprocal Identities

• sin csc
• cos sec
• tan cot

1 csc
1 sin
1 sec
1 cos
1 cot
1 tan
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Signs and Ranges offunction values
in Quadrant sin cos tan cot sec csc
I
II - - - -
III - - - -
IV - - - -
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All Students Take Calculus

Quadrant I (,)
Quadrant II (-,)
xlt0 ygt0 rgt0
xgt0 ygt0 rgt0
All functions are positive
Sin Csc are positive
Tan Cot are positive
Cos Sec are positive
xgt0 ylt0 rgt0
xlt0 ylt0 rgt0
Quadrant III (-,-)
Quadrant IV (,-)
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Ranges for Trig Functions

• For any angle for which the indicated functions
  exist
• -1 lt sin lt 1 and -1 lt cos lt 1
• tan and cot may be equal to any real number
• sec lt -1 or sec gt 1 and
• csc lt -1 or csc gt 1
• (Notice that sec and csc are never between -1
  and 1.)

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The Pythagorean Identities
Remember in a right triangle a2 b2 c2
or using x, y, and r x2 y2 r2 Dividing by
r2
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This is our first trigonometric identity
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(No Transcript)
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Basic trigonometric identities
cos2? sin2? 1 or 1 tan2?
sec2? or tan2? 1 sec2?

cos2?
cos2?
cos2?
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Basic trigonometric identities
cos2? sin2? 1 or cot2? 1
csc2? or 1 cot2? csc2?

sin2?
sin2?
sin2?
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The quotient Identities

• tan
• cot

cos sin