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