1. 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

2. 1.1 Review: Lines, Segments, Rays, AnglesB A
Line 2 distinct points A and B determine a line.
AB Line AB
Segment or Line Segment – the portion of the line between A and B
AB Segment AB A 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.
AB Ray AB A B
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 F P 2
< AFP or <PFA or <F or <2

3. Review of Angles   Acute Angle Less than 90° Right Angle 90°
Obtuse angle
Greater than 90°
0 <  < 90
90 <  < 180
Triangle
180
Circle
360
Straight Angle
180°

4. Special Angle Relationships
Supplementary Angles – Two angles whose sum is 180° 1 2
< 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.

5. Angles and Rotation II I III IV
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
0 degrees
90 degrees
180 degrees
270 degrees II I
Quadrantal Angles lie on the
X or Y axis. (0, 90, 180, 270, 360)
Initial Side
III IV
Counter Clockwise rotation => Positive angle
Clockwise rotation => Negative angle
Coterminal angles have the
same initial and terminal side.

6. 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 = => 12º ’
/60 => º
Convert back to degrees/minutes/seconds
X 60 = => 12º ’
.6333 x 60 = => 12º 42´ 38´´

7. 1.2 Vertical Angles
Vertical angles - non-adjacent angles formed when 2 lines intersect. 1 4 2 3
<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

8. Vertical Angles Theorem
Vertical Angles Theorem: Vertical angles are equal in measure.
145
136
121
Find the missing angles
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.

9. Practice 70° <4 = ________________ <1 = ________________
<2 = ______________
<3 = ________________
70° 4 2
70° 3 1
110°
110° 5x
3x+12 A B
Since vertical angles are congruent, m<ACB = m<DCE? C E D
5x = 3x + 12
<ACB = 5x = 5(6) = 30°
<DCE = 30°
<ACD = 150°
<BCE = 150°
-3x x
2x = 12 2
x = 6

10. Linear Pairs A 7x 2x + 27 B C <ACB and <DCE are supplementary
m<ACB + m<DCE = 180 degrees D E
7x + 2x + 27 = 180
9x + 27 = 180
______________
9x = 153
x = 17
<DCA = 119°
<ACB = 61°
<B CE = 119°
<DCE = 61°

11. Recall From Geometry: Parallel Lines Cut by a Transversal1 2 4 3 5 6 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)

12. 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.

13. 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°)

14. Triangle Angle Sum TheoremB
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)
x = _______________
y = _______________
20° x y
110°

15. Similar Figures (~) Same Shape ABC ~ DEF Not necessarily the same size
Corresponding angles are congruent
Corresponding sides are in proportion D A
ABC ~ DEF
Similarity ratio = 15 = 3 4 5 12 15 E F 3 B C 9

16. 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 6 x =
2.5
100
2.5x = (100)(6)
2.5x = 600
x = 240 feet
personheight
treeheight
personshadow
treeshadow
Similar triangles (proportional sides)
6 = 2.5
.025 = .025
How can you find the hypotenuse & ratios?

17. 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. C A B a b c
Hypotenuse
Pythagorean Theorem
a b2 = c2
(Leg1) (Leg2)2 = (Hypotenuse)2
Legs

18. Special Right Triangles
Isosceles Right Triangle – a triangle with two sides of equal measure. Also called a Triangle.
Triangle 30 60 x 2x
x 3 45 x
x 2

19. 1.3/2.1 Six Trig Functions for Right Triangles
sin () = Opposite csc () = Hypotenuse [cosecant]
Hypotenuse Opposite
cos () = Adjacent sec () = Hypotenuse [secant]
Hypotenuse Adjacent
tan () = Opposite cot() = Adjacent [cotangent]
Adjacent Opposite
Note:
 is an Acute angle.
sin () = 12 csc() = 13
13 12
cos() = 5 sec() = 13
13 5
tan() = 12 / 5 cot() = 5 / 12  13 12  5

20. Trig Functions - Any Angle - Any Quadrant
0 degrees
90 degrees
180 degrees
270 degrees
P(x,y) x y r 
Quadrant I
sin() = opposite/hypotenuse = y/r
cos() =adjacent/hypotenuse = x/r
tan() = opposite/adjacent = y/x
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)
Quadrant IV
sin() = opposite/hypotenuse = -y/r
cos() =adjacent/hypotenuse = x/r
tan() = opposite/adjacent = (-y)/x

21. Trig Functions of Quadrantal Angles
0 degrees
90 degrees
180 degrees
270 degrees
P(0, y)
P(-x, 0)
P(x, 0)
P(0, -y)
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
cos (270) = 0/r = 0
tan (270) = -y/0 = Undefined
sin(180) = 0/r = 0
cos(180) = -x/r = -x/x = -1
tan(180) = 0/(-x) = 0
(See Page 27 for a Complete list for all 6 trigonometric functions.)

22. 1.4 Basic Trig Identities Reciprocal Identities
sin () = 1 cos () = 1 tan () = 1
csc () sec () cot ()
csc () = 1 sec () = 1 cot () = 1
sin () cos () tan ()
Quotient Identities
tan () = sin() cot () = cos()
cos() sin ()
Pythagorean Identities
sin2 () + cos2 () = 1
1 + tan2 () = sec2 ()
1 + cot2 () = csc2 ()