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 B
Line 2 distinct points A and B determine a line.
A
AB Line AB
Segment or Line Segment – the portion of the line between A and B
A
AB Segment AB
B
Ray – part of a line consisting of 1 endpoint , A, and all the points of the line
on 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
B
AB
Ray 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
2
< AFP or <PFA or <F or <2
F
Review of Angles

Right Angle
90°
Straight Angle
180°

Acute Angle
Less than 90°
0 <  < 90
Triangle
180
Obtuse angle
Greater than 90°
90 <  < 180
Circle
360
Special Angle Relationships
Supplementary Angles – Two angles whose sum is 180°
2
1
< 1 and <2 are supplementary
(Remember: A straight angle (line) measures 180°)
Complementary Angles – Two angles whose sum is 90°
<1 and <2 are complementary
1
2
Note: Angles do not have to be adjacent to be supplementary or complementary.
Angles and Rotation
An angle can be thought of as a ‘rotating ray’. The angle’s measure is generated
by 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
Quadrantal Angles lie on the
X or Y axis. (0, 90, 180, 270, 360)
II
90 degrees
I
180 degrees
0 degrees
Initial Side
III
IV
270 degrees
Counter Clockwise rotation => Positive angle
Clockwise rotation => Negative angle
Coterminal angles have the
same initial and terminal side.
Angle Measures
Angles are measured in degrees. (Angles are also measured in
Radians 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 degree
1” (1 Second) = 1/60 of a minute
An angle might measure: 12º 42´ 38´´
Convert to degrees only
38/60 = .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 Angles
Vertical angles - non-adjacent angles formed when 2 lines intersect.
1
4
2
3
<1 and <3 are vertical angles
< 2 and <4 are vertical angles
<1 and <2 are NOT vertical angles
< 3 and <4 are NOT vertical angles
Which of the following are vertical angles?
A.
B.
NOT Vertical
VERTICAL
C.
NOT Vertical
D.
VERTICAL
Vertical Angles Theorem
Vertical Angles Theorem: Vertical angles are equal in measure.
145
Find the missing angles
Step1: Label vertical angle values
136
Step2: Look for linear pairs
121
Step3: Look for complementary angles
Step4: Look for triangles
Step5: Repeat steps 1-4 until all found.
Practice
70°
<4 = ________________
70°
4
3
2
1
70°
<1 = ________________
110°
<2 = ______________
110°
<3 = ________________
A
B Since vertical angles are congruent, m<ACB = m<DCE?
5x
C
D 3x+12 E
5x = 3x + 12
-3x -3x
2x = 12
2
2
x = 6
<ACB = 5x = 5(6) = 30°
<DCE = 30°
<ACD = 150°
<BCE = 150°
A
B
Linear Pairs
C
<ACB and <DCE are supplementary
D
E
m<ACB + m<DCE = 180 degrees
7x + 2x + 27 = 180
9x + 27 = 180
- 27 -27
______________
9x = 153
--------9
9
x
= 17
<DCA = 119°
<ACB = 61°
<B CE = 119°
<DCE = 61°
Recall From Geometry: Parallel Lines
Cut by a Transversal
2
1
4
3
6
5
8
7
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)
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 Theorem
B
A
The sum of the measures of the angles of a
triangle is 180°
m<A + m<B + m<C = 180
C
Practice: (Find the Missing angles)
20°
x y
110°
x = _______________
y = _______________
Similar Figures (~)
 Same Shape
 Not necessarily the same size
 Corresponding angles are congruent
 Corresponding sides are in proportion
D
A
ABC
4
12
5
B
C
9
3
DEF
Similarity ratio = 15 = 3
5 1
15
E
~
F
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?
personheight
treeheight
6
x
= 2.5
100
personshadow
treeshadow
6ft
x
2.5x = (100)(6)
2.5x = 600
2.5
2.5
x = 240 feet
2 ½ ft
100 ft
Similar triangles (proportional sides)
6 = 2.5
240 100
.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.
B
a
C
Legs
c
Hypotenuse
Pythagorean Theorem
a2 + b2 = c2
b
A
(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.
30-60-90 Triangle
30
x 2
45
x
2x
x 3
45
x
60
x
1.3/2.1 Six Trig Functions for Right Triangles
sin () = Opposite
csc () = Hypotenuse
Hypotenuse
Opposite
cos () = Adjacent
sec () = Hypotenuse
Hypotenuse
[secant]
Adjacent
tan () = Opposite
cot() = Adjacent
Adjacent
Opposite
Note:
 is an Acute angle.

12
[cosecant]
[cotangent]
sin () = 12
13
csc() = 13
12
cos() = 5
13
sec() = 13
5
tan() = 12 / 5
cot() = 5 / 12
13

5
Trig Functions - Any Angle - Any Quadrant
90 degrees
P(x,y)
r
180 degrees
Quadrant II
sin() = opposite/hypotenuse = y/r
cos() =adjacent/hypotenuse = -x/r
tan() = opposite/adjacent = y/(-x)
Quadrant III
sin() = opposite/hypotenuse = -y/r
cos() =adjacent/hypotenuse = -x/r
tan() = opposite/adjacent = (-y)/(-x)

x
y
0 degrees
Quadrant I
sin() = opposite/hypotenuse = y/r
270 degrees cos() =adjacent/hypotenuse = x/r
tan() = opposite/adjacent = y/x
Quadrant IV
sin() = opposite/hypotenuse = -y/r
cos() =adjacent/hypotenuse = x/r
tan() = opposite/adjacent = (-y)/x
Trig Functions of Quadrantal Angles
90 degrees
P(0, y)
180 degrees
0 degrees
P(-x, 0)
P(x, 0)
P(0, -y)
270 degrees
sin(90) = y/r = y/y = 1
cos(90) = 0/r = 0
tan(90) = y/0 = Undefined
sin (0) = 0/r = 0
cos (0) = x/r = x/x = 1
tan (0) = 0/x = 0
sin (270) = -y/r = -y/y = -1
sin(180) = 0/r = 0
cos (270) = 0/r = 0
cos(180) = -x/r = -x/x = -1
tan (270) = -y/0 = Undefined
tan(180) = 0/(-x) = 0
(See Page 27 for a Complete list for all 6 trigonometric functions.)
1.4 Basic Trig Identities
Reciprocal Identities
sin () = 1
csc ()
csc () = 1
sin ()
cos () = 1
sec ()
sec () = 1
cos ()
Quotient Identities
tan () = sin()
cos()
Pythagorean Identities
sin2 () + cos2 () = 1
1 + tan2 () = sec2 ()
1 + cot2 () = csc2 ()
cot () = cos()
sin ()
tan () = 1
cot ()
cot () = 1
tan ()