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

Chapter 1Trigonometric Functions

Section 1.1 Basic Concepts

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

The Coordinate Plane

Horizontalx

abscissa

Verticaly

ordinate

Quadrant I

(+,+)

Quadrant II

(-,+)

Quadrant III

(-,-)

Quadrant IV

(+,-)

Pythagorean Theorem

a2 + b2 = c2 hypotenuse

c

leg a

leg

b

C

B

A

Distance Formula

a = (x2 – x1)

b = (y2 – y1)

c = √ a2 + b2

or

distance = √ (x2 – x1)2 + (y2 – y1)2

Midpoint Formula

The Midpoint Formula: The midpoint of a

segment with endpoints (x1 , y1) and (x2 , y2)

has coordinates

Interval Notation

• Set-builder notation{x|x<5} the set of all x such that x is less than 5

• Interval notation(-∞, 5) the set of all x such that x is less than 5

(-∞, 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

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.

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)

Section 1.2 Angles

In this section we will cover:

• Basic terminology

• Degree measure

• Standard position

• Co terminal Angles

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

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

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°

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

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

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.

14

58

23

6

7

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.

14

58

23

6

7

Triangle Properties

The sum of the interior angles of a triangle equal 180°.

Acute – 3 acute anglesRight – 2 acute and one right angleObtuse – 1 obtuse and two acute anglesEquilateral – all sides (and angles) equalIsosceles – two equal sides (and angles)Scalene – no equal sides (or angles)

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.

Section 1.4 Definitions of Trigonometric Functions

In this section we will cover:

• Trigonometric functions– Sine– Cosine– Tangent

• Quadrantal angles

–Cosecant–Secant–Cotangent

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

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 2ñ33

1

sec 2ñ33

ñ2 2 Und

cot ñ3 1 ñ3/3 0

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

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

The Reciprocal Identities

sin £ = csc £ =

cos £ = sec £ =

tan £ = cot £ =

1csc £

1sec £

1cot £

1sin £

1cos £

1tan £

Signs and Ranges offunction values

£ inQuadrant sin £

cos £

tan £ cot £ sec £ csc £

I + + + + + +

II + - - - - +

III - - + + - -

IV - + - - + -

All Students Take Calculus

Quadrant I

(+,+)

Quadrant II

(-,+)

Quadrant III

(-,-)

Quadrant IV

(+,-)

All functions are positive

Sin & Cscare positive

Tan & Cotare positive

Cos & Secare positive

x>0y>0r>0

x<0y>0r>0

x<0y<0r>0

x>0y<0r>0

Ranges for Trig Functions

For any angle £ for which the indicated functions exist:1. -1 < sin £ < 1 and -1 < cos £ < 1;2. tan £ and cot £ may be equal to any

real number;3. sec £ < -1 or sec £ > 1 and

csc £ < -1 or csc £ > 1 (Notice that sec £ and csc £ are never

between -1 and 1.)

The Pythagorean Identities

Remember in a right triangle

a2 + b2 = c2

or using x, y, and r

x2 + y2 = r2

Dividing by r2

x

y

r

x2 + y2 = r2

or

cos2θ + sin2θ = 1

or

sin2θ + cos2θ = 1

r2 r2 r2

x

y

r

θ

This is our first trigonometric identity

cos2θ + sin2θ 1

or

1 + tan2θ = sec2θor

tan2θ + 1 = sec2θ

x

y

r

θ

Basic trigonometric identities

cos2θ =cos2θ cos2θ

cos2θ + sin2θ 1

or

cot2θ + 1 = csc2θor

1 + cot2θ = csc2θ

x

y

r

θ

Basic trigonometric identities

sin2θ sin2θ sin2θ=

The quotient Identities

tan £ = =

cot £ = =

sin £cos £

cos £sin £

yx

xy