angles
DESCRIPTION
Reference book is Emath IV by Oronce et. al.TRANSCRIPT
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Angles and Angle Measure
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Is this an angle? Is this an angle?
Is this an angle?
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Trigonometry
• Trigonometry is the study of triangles and the relationship among their sides and angles.
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T r i g o n o m e t r y
• “gono” – angle
• “tria” - three
• “metria” - measure
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Angles
An angle is formed by two rays, one moving and one stationary that have the same endpoint.
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Concept of Angles in Geometry
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Concept of Angles in Trigonometry
Directed Angles
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2 parts of the angle
1. Initial Side
2. Terminal Side
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2 parts of the angle
1. Initial Side
2. Terminal Side
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2 parts of the angle
1. Initial Side
2. Terminal Side
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Initial Side
- The stationary ray that lies on along the positive x-axis.
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Terminal Side
- The ray that moves clockwise and counter clockwise from the initial side.
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Angle on Standard Position
• An angle is in standard position if its vertex coincides with the origin of the coordinate plane and its initial side coincides with the positive x –axis.
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45°
-315°
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• Positive angles are generated by counterclockwise rotations and negative angles are generated by clockwise rotations.
• Angles are often named by Greek
letters such as α (Alpha), β (Beta), θ(Theta)
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45°
-315° Initial side
Terminalside
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45°
-315°
Coterminal Angles
• Two angles with the same initial and terminal sides
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45°
-315°405°
• There are infinitely many coterminal angles for every given angle.
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To find a coterminal angle,
• Use the formula:
Where:
θ1 is the coterminal angle
θ is the given angle
n is the number of positive or negative revolutions
n3601
revolutions
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Example:
• Find one positive after one revolution and one negative coterminal after 2 revoltuions of 45 degrees.
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45°
-315°
45 °+ 360 ° = 405°
45° + 360°(-2) = - 675°
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Angle Location
• If the terminal side of an angle lies in a given quadrant, then the angle is said to lie in that quadrant.
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α°
“Angle α° lies on the first quadrant” or “Angle α° is located on the first quadrant”
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β°
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θ°
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Angle Location
• If the terminal side of an angle in standard position coincides with a coordinate axis, then the angle is called a quadrantal angle.
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Exercises
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A. Sketch the following angles in standard position. (3 pts. each)
1. -115°
2. 75°
B. Tell the location of each angle. (2 pts. each)
1. 70°
2. 195°
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C. Find the coterminal angles of the ff. by adding two positive and one negative revolution. (2 pts. each)
1. 350° __________ __________
2. - 25° __________ __________
3. 125° __________ __________
4. - 76° __________ __________
5. 80° __________ __________
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Answer key for C
• 1070, -10
• 695, -385
• 845, -235
• 644, -436
• 800, -280
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Assignment:
Note: One revolution = 360°
1. Sketch the angle in standard position: ¾ revolution (5 pts.)
2. Tell the location of the angle in standard position: -(3/5) revolution. (5 points)
3. Bring a protractor tomorrow.