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
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What is Trigonometry? The word trigonometry means “Measurement of Triangles” The study of properties and functions involved in solving triangles. Relationships.
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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.
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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
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)
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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?