Lesson 2 of 5
Convert angles, track rotation, and connect radians to arc length.
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Big ideas · 1 of 8
Big ideas
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A RadianAn angle measured by comparing arc length with radius.Exactly: One radian subtends an arc equal in length to the radius. In general, , where is arc length and is radius.Mix-up alert: Use radians in . Convert a degree measure before using this formula.See this in Key words is the angle subtending an arc equal in length to the radius. A full turn is radians or , so radians and describe the same angle.
For angle conversion, multiply degrees by or radians by . Coterminal angles differ by whole turns: or radians, with integer k.
In standard position, the initial side points along the positive x-axis. Positive rotation is counterclockwise, negative rotation clockwise. Coterminal angles share a terminal side but may describe different amounts of rotation.
The arc length is for a nonnegative angle in radians; for a signed rotation use . For constant rotation through radians in time , angular speed is , and linear speed is radius times angular speed. Keep length and time units consistent.
Step-through
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A circle of radius 9 cm has central angle 80 degrees. Find its arc length exactly.
Whole picture, 4 steps
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Worked example
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Worked example
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Mix-up alert
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Wrong idea: works with when the angle is . Repair: first convert to radians, then multiply by the radius.
Tip
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One radian subtends an arc equal to the radius. Explain why changing the circle size does not change that angle.
Worked example
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For radius and angle :
Watch
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Flashcards
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A semicircle is radians.
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How many radians are in one full turn?
.
Convert to radians.
.
Convert to degrees.
.
Which direction is positive angle rotation?
Counterclockwise.
What makes two angles coterminal?
They differ by an integer number of full turns.
State the arc-length formula and its angle unit.
for nonnegative in radians; for signed rotation, arc length is .
What is linear speed for radius and angular speed ?
, with in radians per time unit.
Do coterminal angles always describe equal rotation amounts?
No. They can differ by complete turns.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Write it
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Write it
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Match
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Connect the ideas
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