1. DMS, Arc Length, and Sector Area
Section 4.1A
Precalculus PreAP/Dual, Revised ©2016
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4.1A: DMS, Arc Length, and Sector Area

2. Degrees Minutes Seconds (DMS)
Written as: 𝑫° 𝑴′ 𝑺′′
It can also be written in decimal degree form or degree form
Make sure it is in degree mode on the calculator
Boaters use DMS to help track other vestals
Pilots use DMS to track oncoming planes or assist to land
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4.1A: DMS, Arc Length, and Sector Area

3. Steps in Writing in Decimal Form
Keep the first digits in degree form
Label the second number over 60 (how many minutes are there in an hour?) and convert the second number into a decimal form from a fraction form
Label the third number over 3600 (how many seconds are there in an hour?)
Add the digits together and label as degrees
If the given is negative, exclude the negative until the end
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4.1A: DMS, Arc Length, and Sector Area

4. 4.1A: DMS, Arc Length, and Sector Area
Example 1
Convert 𝟑𝟔° 𝟏𝟒′ 𝟐𝟗′′ to decimal form. Round to 4 decimal places. Step 1: Keep the first digits in degree form
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4.1A: DMS, Arc Length, and Sector Area

5. 4.1A: DMS, Arc Length, and Sector Area
Example 1
Convert 𝟑𝟔° 𝟏𝟒′ 𝟐𝟗′′ to decimal form. Round to 4 decimal places. Step 2: Label the second number over 60 (how many minutes are there in a hour?) and convert into a decimal
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4.1A: DMS, Arc Length, and Sector Area

6. 4.1A: DMS, Arc Length, and Sector Area
Example 1
Convert 𝟑𝟔° 𝟏𝟒′ 𝟐𝟗′′ to decimal form. Round to 4 decimal places. Step 3: Label the third number over 3600 (how many seconds are there in a hour?) using the calculator and convert it to a decimal
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4.1A: DMS, Arc Length, and Sector Area

7. 4.1A: DMS, Arc Length, and Sector Area
Example 1
Convert 𝟑𝟔° 𝟏𝟒′ 𝟐𝟗′′ to decimal form. Round to 4 decimal places. Step 4: Add the figures together
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4.1A: DMS, Arc Length, and Sector Area

8. 4.1A: DMS, Arc Length, and Sector Area
Example 2
Convert −𝟓° 𝟓𝟕′ 𝟑𝟔′′ to decimal form. Round to 4 decimal places.
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4.1A: DMS, Arc Length, and Sector Area

9. 4.1A: DMS, Arc Length, and Sector Area
Your Turn
Convert 𝟑𝟓° 𝟏𝟓′ 𝟐𝟕′′ to decimal form. Round to 4 decimal places.
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4.1A: DMS, Arc Length, and Sector Area

10. Steps in Writing in Degree Form
Keep the first digits in degree form
Multiply the last numbers with the decimal (behind the degrees) by 60
Take the decimals from the previous answer in step 2 and multiply by 60 again
Put them together and label accordingly
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4.1A: DMS, Arc Length, and Sector Area

11. 4.1A: DMS, Arc Length, and Sector Area
Example 3
Convert 𝟒𝟖.𝟑𝟔𝟐𝟓° to DMS form. Step 1: Keep the first digits in degree form Step 2: Multiply the last numbers with the decimal (behind the degrees) by .6 (60 degrees/1 minute)
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4.1A: DMS, Arc Length, and Sector Area

12. 4.1A: DMS, Arc Length, and Sector Area
Example 3
Convert 𝟒𝟖.𝟑𝟔𝟐𝟓° to DMS form. Step 3: Take the decimals from the previous answer in step 2 and multiply by 60 again (60 seconds/1 minute)
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4.1A: DMS, Arc Length, and Sector Area

13. 4.1A: DMS, Arc Length, and Sector Area
Example 4
Convert 𝟏𝟑.𝟏𝟐𝟑𝟒𝟓° to DMS form.
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4.1A: DMS, Arc Length, and Sector Area

14. 4.1A: DMS, Arc Length, and Sector Area
Your Turn
Convert 𝟒𝟑.𝟓𝟓𝟐𝟓° to DMS form.
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4.1A: DMS, Arc Length, and Sector Area

15. 4.1A: DMS, Arc Length, and Sector Area
How do we find the arc length? What is circumference of a circle? 𝑻 𝑺 𝑹
central
angle
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4.1A: DMS, Arc Length, and Sector Area

16. 4.1A: DMS, Arc Length, and Sector Area
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=𝒓𝜽
Degrees must be converted to Radians
Do NOT forget the units
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4.1A: DMS, Arc Length, and Sector Area

17. 4.1A: DMS, Arc Length, and Sector Area
Example 5
Determine the Arc Length of a circle with the given radius of 𝒓=𝟒 inches and 𝜽= 𝝅 𝟔 .
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4.1A: DMS, Arc Length, and Sector Area

18. 4.1A: DMS, Arc Length, and Sector Area
Example 6
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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4.1A: DMS, Arc Length, and Sector Area

19. 4.1A: DMS, Arc Length, and Sector Area
Example 6
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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4.1A: DMS, Arc Length, and Sector Area

20. 4.1A: DMS, Arc Length, and Sector Area
Your Turn
Determine the Arc Length of a circle with the given radius of 𝒓=𝟒 inches and 𝜽=𝟐𝟒𝟎°. Make sure to convert degrees to radians.
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4.1A: DMS, Arc Length, and Sector Area

21. 4.1A: Arc Length and Linear and Angular Speed
Area of a Sector Area
Sector is a circle of the region bounded by two radii of the circle and their intercepted arc
Equation: 𝑨= 𝟏 𝟐 𝒓 𝟐 𝜽 where radius is the base and 𝜽 is the height
Always leave answers in radian mode
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4.1A: Arc Length and Linear and Angular Speed

22. 4.1A: Arc Length and Linear and Angular Speed
Example 7
A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates through an angle of 𝟏𝟐𝟎°. Find the area of the fairway watered by the sprinkler.
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4.1A: Arc Length and Linear and Angular Speed

23. 4.1A: Arc Length and Linear and Angular Speed
Example 7
A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates through an angle of 𝟏𝟐𝟎°. Find the area of the fairway watered by the sprinkler.
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4.1A: Arc Length and Linear and Angular Speed

24. 4.1A: Arc Length and Linear and Angular Speed
Example 8
A sector of a circle of radius 24 m has an area of 288 𝒎². Find the central angle of this sector.
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4.1A: Arc Length and Linear and Angular Speed

25. 4.1A: Arc Length and Linear and Angular Speed
Example 9
Three circles with radii 1, 2, and 3 feet are externally tangent to one another, as shown in the figure. Find the area of the sector of the circle of radius 1 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.
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4.1A: Arc Length and Linear and Angular Speed

26. 4.1A: Arc Length and Linear and Angular Speed
Your Turn
A sector of a circle has a central angle of 𝟔𝟎°. Find the area of the sector if the radius of the circle is 3 miles.
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4.1A: Arc Length and Linear and Angular Speed

27. 4.1A: DMS, Arc Length, and Sector Area
Assignment
Page all, 76
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4.1A: DMS, Arc Length, and Sector Area