1. 6.3A: Coterminal and Arc Lengths
Page 441
11)
36°
13)
–18°
15)
135°
17) 4°
19)
–75°
21)
972°
23)
π/30
25)
–π/15
27)
5π/12
29)
3π/4
31)
–5π/4
33)
31π/6
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6.3A: Coterminal and Arc Lengths

2. 6.3A: Coterminal and Arc Lengths
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6.3A: Coterminal and Arc Lengths

3. Co-Terminal and Arc Lengths
Section 6.3
Pre-Calculus AB Pre-AP/Dual, Revised ©2014
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6.3A: Coterminal and Arc Lengths

4. 6.3A: Coterminal and Arc Lengths
Co-Terminal Angles
Coterminal angles are angles in standard position with the same terminal side
To determine the coterminal angles, add and/or subtract 360° by rotating counter clockwise for a positive rotation
Initial Ray is the positive x-axis
Terminal Ray is the location where the ray ends
Coterminal Angles can be negative
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6.3A: Coterminal and Arc Lengths

5. 6.3A: Coterminal and Arc Lengths
Example 1
Find two co-terminal rays (1 positive and 1 negative) of 40°
40°
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6.3A: Coterminal and Arc Lengths

6. 6.3A: Coterminal and Arc Lengths
Example 2
Find one positive and one negative co-terminal ray of 𝟕𝝅 𝟑 .
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6.3A: Coterminal and Arc Lengths

7. 6.3A: Coterminal and Arc Lengths
Example 3
Find one positive and one negative co-terminal ray of 𝟐𝟗𝝅 𝟔 .
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6.3A: Coterminal and Arc Lengths

8. 6.3A: Coterminal and Arc Lengths
Your Turn
Find one positive and one negative co-terminal ray of − 𝟕𝝅 𝟗
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6.3A: Coterminal and Arc Lengths

9. 6.3A: Coterminal and Arc Lengths
Sector is a region of the circle that bounded by two radii and an arc of a circle
The Central Angle of a sector is the angle formed by the two radii
Arc Length equation: s = rө
Degrees must be converted to Radians
Do NOT forget the units
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6.3A: Coterminal and Arc Lengths

10. 6.3A: Coterminal and Arc Lengths
Arc Length equation: s = rө T S R
central
angle
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6.3A: Coterminal and Arc Lengths

11. 6.3A: Coterminal and Arc Lengths
Example 4
Determine the Arc Length with the given radius of r = 4 inches and ө = π/6.
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6.3A: Coterminal and Arc Lengths

12. 6.3A: Coterminal and Arc Lengths
Example 5
The second hand on a clock is 6 inches long. How far does the tip of the second hand move in 15 seconds? 1 full rotation is 60 seconds. Round to 4 decimal places.
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6.3A: Coterminal and Arc Lengths

13. 6.3A: Coterminal and Arc Lengths
Example 6
The radius is 36 cm. Find the angle when the arc length is 5 meters. Leave answer in Radian mode.
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6.3A: Coterminal and Arc Lengths

14. 6.3A: Coterminal and Arc Lengths
Your Turn
The second hand on a clock is 6 inches long. How far does the tip of the second hand move in 1 minute and 10 seconds? Hint: Keep the answer in seconds.
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6.3A: Coterminal and Arc Lengths

15. Linear and Angular Speed
Angular Speed applies to any object or particle that turns; angle through with the point rotates over time (also known as angle rotation)
Angular Speed Equation: 𝑨𝒏𝒈𝒍𝒆 𝑻𝒊𝒎𝒆 =𝝎= 𝜽 𝒕
Linear speed applies to any object or particle that moves; distance that the point travels over time (distance)
Linear Speed Equation: 𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝑻𝒊𝒎𝒆 =𝑽=𝒓𝝎 where 𝜽 𝒕 is the angular speed
Therefore, Linear Speed is also known as (Radius) * (Angular Speed)
The angular speed of an object traveling in a circular path is the same, regardless of its distance from the center of the circle. When the angular speed of the object stays the same, the linear speed increases as the object moves farther from the center
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6.3A: Coterminal and Arc Lengths

16. Angular Speed (Rotation)
Distance = Rate • Time
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6.3A: Coterminal and Arc Lengths

17. Linear Speed (Distance)
Distance = Rate • Time
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6.3A: Coterminal and Arc Lengths

18. 6.3A: Coterminal and Arc Lengths
Example 7
A merry go round makes 8 revolutions feet per minute. What is the angular speed?
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6.3A: Coterminal and Arc Lengths

19. 6.3A: Coterminal and Arc Lengths
Example 8
A merry go round makes 8 revolutions per minute. The horse 12 feet from the center is traveling with a radius of 12. How fast is the horse going per hour?
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6.3A: Coterminal and Arc Lengths

20. 6.3A: Coterminal and Arc Lengths
Example 9
An earth satellite in circular orbit 1200 km high makes one complete revolution every 90 min. What is its linear speed? Use 6400 km for the length of a radius of the earth.
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6.3A: Coterminal and Arc Lengths

21. 6.3A: Coterminal and Arc Lengths
Your Turn
A circular mobile above a baby’s crib makes 5 revolutions per minute. What is the angular speed of the mobile in radians per minute and how fast is a toy 9 inches from the center traveling in feet per minute?
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6.3A: Coterminal and Arc Lengths

22. 6.3A: Coterminal and Arc Lengths
Assignment
Page odd 35-41: Disregard the 0 to 2π. List one positive and one negative coterminal angle. 43, 45: 2 coterminal angles is fine, one positive and one negative
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6.3A: Coterminal and Arc Lengths