What is Trigonometry?The word trigonometry means “Measurement of Triangles”

• The study of properties and functions involved in solving triangles.

• Relationships among sides and angles of triangles

•Phenomena that occur in cycles and/or waves, rotations & vibrations

APPLICATIONS

Astronomy Planetary Orbits

Navigation Light Rays

Surveying Sound Waves DNA Research

1.1 Review: Lines, Segments, Rays, Angles

Line 2 distinct points A and B determine a line.

Segment or Line Segment – the portion of the line between A and B

A B AB Segment AB

Ray – part of a line consisting of 1 endpoint , A, and all the points of the lineon 1 side of the endpoint. -- in other words, the portion of line AB that starts at A& continues through B and on past B, is called Ray AB.

A BAB Ray AB

A

B

AB Line AB

Angle – 2 rays or 2 line segments with a common endpoint. The rays are the sides of the angle & the common endpoint is the vertex. A

FP

< AFP or <PFA or <F or <2 2

Review of Angles

Right Angle 90°

Acute AngleLess than 90°

Obtuse angleGreater than 90°

0 < < 90 90 < < 180

Straight Angle 180°

Triangle 180

Circle 360

Special Angle Relationships

Supplementary Angles – Two angles whose sum is 180°

< 1 and <2 are supplementary(Remember: A straight angle (line) measures 180°)

12

Complementary Angles – Two angles whose sum is 90°

<1 and <2 are complementary

Note: Angles do not have to be adjacent to be supplementary or complementary.

12

Angles and RotationAn angle can be thought of as a ‘rotating ray’. The angle’s measure is generatedby a rotation about the angle’s vertex, from the initial side to the terminal side. An angle is in standard position if • the vertex is at the origin of the x/y axes and

• the initial side of the angle lies along the positive x-axis

0 degrees

90 degrees

180 degrees

270 degrees

Initial Side

Counter Clockwise rotation => Positive angleClockwise rotation => Negative angle

III

III IV

Coterminal angles have thesame initial and terminal side.

Quadrantal Angles lie on theX or Y axis. (0, 90, 180, 270, 360)

Angle MeasuresAngles are measured in degrees. (Angles are also measured inRadians which we will discuss later)

One complete rotation of a ray (forming an angle) is 360º

Minutes and Seconds measure portions of a degree.1’ (1 Minute) = 1/60 of a degree1” (1 Second) = 1/60 of a minute

An angle might measure: 12º 42´ 38´´Convert to degrees only38/60 = .6333 => 12º 42.6333’42.6333/60 => 12.710555º

Convert back to degrees/minutes/seconds.710555 X 60 = 42.6333 => 12º 42.6333’.6333 x 60 = 37.998 => 12º 42´ 38´´

1.2 Vertical AnglesVertical angles - non-adjacent angles formed when 2 lines intersect.

1

3

24

<1 and <3 are vertical angles <1 and <2 are NOT vertical angles< 2 and <4 are vertical angles < 3 and <4 are NOT vertical angles

Which of the following are vertical angles?

A. B. C. D.

NOT Vertical VERTICAL NOT Vertical VERTICAL

Vertical Angles Theorem

Vertical Angles Theorem: Vertical angles are equal in measure.

145

Find the missing angles

136

121

Step1: Label vertical angle values

Step2: Look for linear pairs

Step3: Look for complementary angles

Step4: Look for triangles

Step5: Repeat steps 1-4 until all found.

Practice

42

13

<4 = ________________

<1 = ________________

<2 = ______________

<3 = ________________

70°

5x

3x+12

C

A B

D E

Since vertical angles are congruent, m<ACB = m<DCE?

5x = 3x + 12

-3x -3x

2x = 12

2 2

x = 6

<ACB = 5x = 5(6) = 30°<DCE = 30° <ACD = 150°<BCE = 150°

70°

70°

110°

110°

7x2x

+ 27

<ACB and <DCE are supplementary

m<ACB + m<DCE = 180 degrees

A B

C

DE

7x + 2x + 27 = 180

9x + 27 = 180 - 27 -27______________ 9x = 153 ----- ----- 9 9

x = 17

<DCA = 119°<ACB = 61°<B CE = 119°<DCE = 61°

Linear Pairs

Recall From Geometry: Parallel Lines Cut by a Transversal

If two parallel lines are cut by a transversal then • Corresponding angles are congruent. (Ex: <2 and <6)

• Alternate interior angles are congruent (Ex: <3 and <5)• Alternate exterior angles are congruent (Ex: <1 and <7)

• Same side interior angles are supplementary (Ex: <3 and <6)• Same side exterior angles are supplementary (Ex: <2 and <7)

12

4 3

5 6

8 7

Review of Triangles

Triangle – 3 sided closed figure where all sides are line segments connected at their endpoints.

Classifying Triangles by SIDES

Equilateral Triangle – A triangle with all 3 sides equal in measure.

Isosceles Triangle – A triangle in which at least 2 sides have equal measure.

Scalene Triangle – A triangle with all 3 sides of different measure.

Classifying Triangles by Angle

Right Triangle – A triangle that has a 90 angle

Obtuse Triangle – A triangle with an obtuse angle (greater than 90)

Acute Triangle – A triangle with ALL angles less than 90

Equiangular Triangle – A triangle with all angles of equal measure. (All angles will measure 60°)

Triangle Angle Sum TheoremA B

C

The sum of the measures of the angles of a triangle is 180°

m<A + m<B + m<C = 180

Practice: (Find the Missing angles)

20°

110°

x y x = _______________

y = _______________

Similar Figures (~) Same Shape

Not necessarily the same size

Corresponding angles are congruent

Corresponding sides are in proportion

9

12 15

3

54

A

B C

E F

D

ABC ~ DEF

Similarity ratio = 15 = 3 5 1

The Shadow Problem (Using Similar Triangles)

Juan is 6 feet tall, but his shadow is only 2 ½ feet long.There is a tree across the street with a shadow of 100 feet.The sun hits the tree and Juan at the same angle to make the shadows.How tall is the tree?

6ft

2 ½ ft

x

100 ft

6x

= 2.5100

2.5x = (100)(6)

2.5x = 6002.5 2.5

x = 240 feet

personheight treeheight

personshadow

treeshadow

Similar triangles (proportional sides) 6 = 2.5240 100 .025 = .025

How can you find the hypotenuse & ratios?

Pythagorean Theorem (for Right Triangles)

Right Angle – An angle with a measure of 90°

Right Triangle – A triangle that has a right angle in its interior.

Legs

Hypotenuse

C A

B

a

b

cPythagorean Theorem

a2 + b2 = c2

(Leg1)2 + (Leg2)2 = (Hypotenuse)2

Special Right Triangles

Isosceles Right Triangle – a triangle with two sides of equal measure. Also called a 45-45-90 Triangle.

45

45

x

x

x 2

30

60

x

2xx 3

30-60-90 Triangle

1.3/2.1 Six Trig Functions for Right Trianglessin () = Opposite csc () = Hypotenuse [cosecant]

Hypotenuse Opposite

cos () = Adjacent sec () = Hypotenuse [secant]

Hypotenuse Adjacent

tan () = Opposite cot() = Adjacent [cotangent]

Adjacent Opposite

5

1213

sin () = 12 csc() = 13 13 12

cos() = 5 sec() = 13 13 5

tan() = 12 / 5 cot() = 5 / 12

Note: is an Acute angle.

Trig Functions - Any Angle - Any Quadrant

0 degrees

90 degrees

180 degrees

270 degrees

x

y

P(x,y)r

Quadrant I sin() = opposite/hypotenuse = y/rcos() =adjacent/hypotenuse = x/rtan() = opposite/adjacent = y/x

Quadrant II sin() = opposite/hypotenuse = y/rcos() =adjacent/hypotenuse = -x/rtan() = opposite/adjacent = y/(-x)

Quadrant IV sin() = opposite/hypotenuse = -y/rcos() =adjacent/hypotenuse = x/rtan() = opposite/adjacent = (-y)/x

Quadrant III sin() = opposite/hypotenuse = -y/rcos() =adjacent/hypotenuse = -x/rtan() = opposite/adjacent = (-y)/(-x)

Trig Functions of Quadrantal Angles

0 degrees

90 degrees

180 degrees

270 degrees

P(x, 0)

sin (0) = 0/r = 0cos (0) = x/r = x/x = 1tan (0) = 0/x = 0

P(-x, 0)

sin(180) = 0/r = 0cos(180) = -x/r = -x/x = -1tan(180) = 0/(-x) = 0 (See Page 27 for a Complete list for all 6 trigonometric functions.)

P(0, y)

sin(90) = y/r = y/y = 1cos(90) = 0/r = 0tan(90) = y/0 = Undefined

sin (270) = -y/r = -y/y = -1cos (270) = 0/r = 0tan (270) = -y/0 = Undefined

P(0, -y)

1.4 Basic Trig Identities

sin () = 1 cos () = 1 tan () = 1

csc () sec () cot ()

csc () = 1 sec () = 1 cot () = 1

sin () cos () tan ()

tan () = sin() cot () = cos() cos() sin ()

sin2 () + cos2 () = 11 + tan2 () = sec2 ()1 + cot2 () = csc2 ()

Reciprocal Identities

Quotient Identities

Pythagorean Identities