angles and their measures
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Angles and their Measures. Lesson 1. As derived from the Greek Language, the word trigonometry means “measurement of triangles.” Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying. - PowerPoint PPT PresentationTRANSCRIPT
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Angles and their Measures
Lesson 1
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As derived from the Greek Language, the word trigonometry means “measurement of triangles.”
Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying.
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With the development of Calculus and the physical sciences in the 17th Century, a different perspective arose – one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domain.
Consequently the applications expanded to include physical phenomena involving rotations and vibrations, including sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.
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We will explore both perspectives beginning with angles and their measures…..
An angle is determined by rotating a ray about its endpoint.
The starting position of called the initial side. The ending position is called the terminal side.
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Standard PositionVertex is at the origin, and the initial side is on the x-axis.
Initial Side
Termin
al Sid
e
360,0
90
180
270
III
III IV
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Positive Angles are generated by counterclockwise rotation.
Negative Angles are generated by clockwise rotation.
Let’s take a look at how negative angles are labeled on the coordinate graph.
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Negative AnglesGo in a Clockwise rotation
45
360,0
90
180
270
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Coterminal Angles
Angles that have the same initial and terminal side. See the examples below.
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Coterminal AnglesThey have the same initial and terminal sides.
Determine 2 coterminal angles, one positive and one negative for a 60 degree angle.
60
60 + 360 = 420 degrees
60 – 360 = -300 degrees
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Give 2 coterminal angles.
30
30 + 360 = 390 degrees
30 – 360 = -330 degrees
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Give a coterminal angle, one positive and one negative.
230
230 + 360 = 590 degrees
230 – 360 = -130 degrees
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Give a coterminal angle, one positive and one negative.
20
-20 + 360 = 340 degrees
-20 – 360 = -380 degrees
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Give a coterminal angle, one positive and one negative.
460 + 360 = 820 degrees
460 – 360 = 100 degrees100 – 360 = -260 degrees
460
Good but not bestanswer.
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Complementary AnglesSum of the angles is 90
40 120
Find the complement of each angles:
40 + x = 90
x = 50 degrees
No Complement!
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Supplementary AnglesSum of the angles is 180
40 120
Find the supplement of each angles:
40 + x = 180
x = 140 degrees
120 + x = 180
x = 60 degrees
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Coterminal Angles:
To find a Complementary Angle:
To find a Supplementary Angle:
360Angle
90 Angle
180 Angle
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Radian Measure
One radian is the measure of the central angle, , that intercepts an arc, s, that is equal in length to the radius r of the circle.
So…1 revolution is equal to 2π radians
2C r
2
2
2
2C
r
rC
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902
180
3602
radians
radians
radians
Let’s take alook at themplaced on theunit circle.
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Radians
Now, let’s add more…..
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Radians
22
1
radians4
radians4
3
24
radians4
5radians
4
7
rad28.6 rad14.3
rad57.1
rad71.4
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More Common Angles
Let’s take a look at more common angles that are found in the unit circle.
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Radians
radians4
radians4
3
radians4
5 radians4
7
rad28.6 rad14.3
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Radians
radians4
radians4
3
radians4
5 radians4
7
rad28.6 rad14.3
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Look at the Quadrants
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Determine the Quadrant of the terminal side of each given angle.
37
122
3371
Q1
Go a little more than one quadrant – negative. Q3
A little more than one revolution. Q1
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Determine the Quadrant of the terminal side of each given angle.
14
5156
9
8240
1000
Q3
Q2
2 Rev + 280 degrees. Q4
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Coterminal Angles using Radians
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Find a coterminal angle.
There are an infinite number of coterminal angles!
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Give a coterminal angle, one positive and one negative.
13
4
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Give a coterminal angle, one positive and one negative.
5
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2
5
Find the complement of each angles:
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2
5
Find the supplement of each angles:
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2
3
Find the complement & supplement of each angles, if possible:
None
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Coterminal Angles:
To find a Complementary Angle:
To find a Supplementary Angle:
2Angle
2Angle
Angle
RECAP
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Conversions
radians180
1
1801 radian
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4
3
180
135
180135
x
NOTE: The answer is in radians!
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2
3
180
270
180270
x
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810180
2
9
x
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Convert 2 radians to degrees
59.114360180
2
x
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The relationship between arc length, radius, and central angle is
Arc Length
Arc Length = (radius) (angle)
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1st Change 240 degrees into radians.
radiansx3
4
180
240
180240
inchesors
s
76.163
16
3
44