Angles, Arcs,

1
Angles, Arcs, Similar Triangles

• Marguerite Smith
• Merced College

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What is Trigonometry?

• The word Trigonometry literally means
  measurement of triangles
• The relationships among the sides and angles of
  triangles are used in surveying, navigation, and
  astronomy
• Trigonometric relationships are also functions
  that apply to physical phenomena such as sound
  waves, light rays, vibrating strings, pendulums,
  planetary orbits and orbits of atomic particles

3
What is an ANGLE in Trigonometry?

• An angle is formed by rotating a half-line,
  called a ray, around its end point. One ray is
  fixed, and is called the initial side.
• The second ray is called the terminal side.
• The common end point is called the vertex.

Terminal Side
q
Vertex
Initial Side
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ANGLES on a Rectangular Coordinate System

• The Vertex is placed at the Origin
• The Initial side is always the positive X-axis. .

Y
X
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Positive, Negative Coterminal Angles

• A positive angle results from a counter-clockwise
  rotation.

• A negative angle results from a clockwise
  rotation.

135 º
-225 º
Two angles with the same terminal side are
coterminal. So 135 º and - 225 º are
coterminal!
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Degree Measure of Angles

• An angle formed by one complete revolution has a
  measure of 360 degrees.
• An angle of 1 degree is 1/360th of a complete
  revolution.
• A 90 º angle is called a right angle

• A 180 º angle forms a straight line and is called
  a straight angle

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Right, Acute, and Obtuse Angles
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Degrees, Minutes and Seconds

• A degree is 1/360th of a complete circle
• A minute is 1/60th of a degree
• A second is 1/3600th of a degree
• (Also, a second is 1/60th of a minute)

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From DMS to DD

• On a calculator, DMS means degrees, minutes and
  seconds.
• DD means decimal degree form.
• On a scientific calculator, enterto get the
  decimal degree form, 12.106 º
• Check your users manual for a shorter way to
  make DMS to DD conversions.

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DMS to DD conversions

• On a TI-86 graphing calculator, press 2nd MATH,
  then ANGLE
• Press 12 6 23 ENTER (Use the F3 key for
  the )

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Complementary Supplementary Angles
TWO angles that have a sum of 90º are
complementary


TWO angles that have a sum of 180º are
supplementary


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Arcs and Central Angles
The length of arc s can be found by the
proportion
q s
C
Where C is the Circumference of the circle and q
is the Central Angle subtending arc s.
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Similar Triangles
a b
a b
c
c
If two triangles are similar, their corresponding
sides are proportional
Triangles are similar if two corresponding angles
are equal.
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Similar Triangles
a b
a b
c
c
If two triangles are similar, then it is also
true that
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Special Triangles 45º- 45º- 90º
1
1
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All 45º- 45º- 90º Triangles are Similar!
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Special Triangles 30º- 60º- 90º
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All 30º- 60º- 90º Triangles are Similar!
4
2
1
½
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All 30º- 60º- 90º Triangles are Similar!
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Acknowledgements

• This presentation was made possible by training
  and equipment provided by an Access to Technology
  grant from Merced College.
• Textbooks consulted were
• Trigonometry Fourth Edition by Larson Hostetler
• Analytic Trigonometry with Applications Seventh
  Edition by Barnett, Ziegler Byleen