MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions

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MTH 112Elementary Functions Chapter 5The
Trigonometric Functions

• Section 4 Radians, Arc Length, and Angular Speed

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Angle Measurement Degrees

• Full rotation from the positive x-axis to the
  positive x-axis is 360.
• Why?
• Therefore, rotation from the positive x-axis to
• the positive y-axis is 90.
• the negative x-axis is 180.
• the negative y-axis is 270.

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Problem Find the length of an arc of a circle of
radius r that is intercepted by a central angle
of degree measure ?.
Circumference of the circle is 2?r.
The arc is ?/360th of the circle.
Therefore, a is ?/360th of 2?r.
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Angle Measurement Radians

• The measure of the angle ? in radians is the
  length of the arc intercepted by the angle,
  divided by the radius.
• ? a / r radians

r
a
?
r
x2 y2 r2
Since any circle produces the same results, a
unit circle is used (i.e. r 1).
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Angle Measurement Radians vs.
Degrees

• With a right angle
• 90 ? / 2 radians
• ? 1? ? / 180 radians
• ? a (? / 180)a radians
• ? b radians (180 / ?)b

Circumference of a Unit Circle 2?
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Angle Measurement Radians vs.
Degrees

• Degree and DMS measurements are designated using
  the symbols , , and .
• Examples
• 25.47
• 252812
• Radian measurements do not use any symbols.
• Examples
• .4445353605
• 283? / 2000

NOTE All of the examples above are the same
measurement (third one is approximate).
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Common Angles Degrees vs. Radians
First Quadrant
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Common Angles Degrees vs. Radians
Second Quadrant
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Common Angles Degrees vs. Radians
Third Quadrant
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Common Angles Degrees vs. Radians
Fourth Quadrant
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The Unit Circle - Summarized
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The Unit Circle Summarized Axis Points
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The Unit Circle Summarized k?/6 Points
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The Unit Circle Summarized k?/4 Points
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The Unit Circle Summarized k?/3 Points
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Other Angles

• Negative Angles
• Go clockwise (e.g. -?/6 is the same point as
  11?/6)
• Ordered pairs are the same
• Angles Greater than 2?
• Repeats the same points
• General Rule
• Add or subtract multiples of 2? to get the
  radian measure between 0 and 2?.

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Problem Find the length of an arc of a circle of
radius r that is intercepted by a central angle
of radian measure ?.
Circumference of the circle is 2?r.
The arc is ?/(2?)th of the circle.
Therefore, a is ?/(2?)th of 2?r.