objectives: 1. use a rotating ray to extend the definition of angle measure to negative angles and...
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OBJECTIVES :
1. USE A ROTATING RAY TO EXTEND THE DEFINITION OF ANGLE MEASURE TO NEGATIVE ANGLES AND ANGLES GREATER THAN 180° .
2. DEFINE RADIAN MEASURE AND CONVERT ANGLE MEASURES BETWEEN DEGREES AND RADIANS.
6.3Angles & Radian Measure
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Angles of Rotation
Positive angles are rotated counter-clockwise & negative angles clockwise.
Standard position has the initial side on the x-axis & the vertex on the origin.
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Radians & the Unit Circle
Radians are used to measure angles using arc length.
Circumference:
1801or
180radian 1
180
3602)1(22
rC
r = 1
0° = 360° = 2π
290
180° = π
2
3270
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Example #1Convert from Radians to Degrees
A.
B.
C.
9
5
3
5
20180
108180
900180
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Example #2Convert from Degrees to Radians
A. 150°
B. -330°
C. 540°
6
5
180
6
11
180
3
180
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Example #3Find the angle measures from each graph.
360° - 60°= 300°
-360° + 90° + 115° = -155°
5(180°) = 900°
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Example #4Draw the following angles in standard position. State the quadrant in which the terminal side is located.
A. -110°
B. 530°
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Example #4Draw the following angles in standard position. State the quadrant in which the terminal side is located.
C. 3400°
D. 3
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Example #4 (continued…)Draw the following angles in standard position. State the quadrant in which the terminal side is located.
E.
F.
3
4
3
7
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Arc Length of a Circle
Depending on whether an angle is given in radians or degrees the formulas for arc length vary slightly, although the concept remains the same.
For radians:
For degrees:
rL
rL
180
The key to learning this is not to memorize either formula, but to build on what you already know. The length of an arc is a fraction of the distance around the entire circle (circumference).
Multiply that fraction by the circumference of the circle and you get the arc length.
rwhole
partL
ncecircumferewhole
partL
2
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Sector Area of a Circle
Depending on whether an angle is given in radians or degrees the formulas for sector area also vary.
For radians:
For degrees: 2
360rA
2
2rA
And just like arc length, the formulas for sector area are based on the same concept:
2rwhole
partA
areawhole
partA
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Example #5Find the Arc Length & Sector Area of the following:
A.
2
2
2
cm 25.205
3
196196
3
1
1962
1
3
2
1423
2
rwhole
partA
cm 32.29
3
2828
3
1
282
1
3
2
14223
2
2
rwhole
partL
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Example #5Find the Arc Length & Sector Area of the following:
B.
2
2
2
cm 66.125
40
14418
5
12360
100
rwhole
partA
ft 94.203
20
2418
5
122360
100
2
rwhole
partL
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The second hand on a clock is 5 inches long. How far does the tip of the hand move in 45 seconds?
12
6
39
1
2
4
578
10
11
Example #6Arc Length
5’’
in6.23
2
1510
4
3
5260
45
2
rwhole
partL