Math Exercises & Math Problems: Angle Unit Conversion - Degrees ... - Free Printable

It looks like you've uploaded a list of angle conversions between degrees, radians, and minutes/seconds. The task is to convert each given angle from one unit to another as indicated.

Let’s go through each item step by step, converting them appropriately using standard conversion formulas:

Key Conversion Formulas:

- $ 180^\circ = \pi \text{ rad} $
- $ 1^\circ = 60' $ (minutes)
- $ 1' = 60'' $ (seconds)
- So:
- To convert degrees to radians: $ \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} $
- To convert radians to degrees: $ \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi} $
- To convert minutes to degrees: $ \theta_{\text{deg}} = \frac{\theta_{\text{min}}}{60} $

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We'll solve each part:

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a) $ 180^\circ $ → (rad)
$ 180^\circ = \pi $ rad
✔ Answer: $ \pi $ rad

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b) $ 3\pi $ → (°)
$ 3\pi \times \frac{180}{\pi} = 3 \times 180 = 540^\circ $
✔ Answer: $ 540^\circ $

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c) $ 7,200' $ → (rad)
First convert minutes to degrees:
$ 7,200' = \frac{7200}{60} = 120^\circ $
Now convert to radians:
$ 120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3} $ rad
✔ Answer: $ \frac{2\pi}{3} $ rad

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d) $ 15^\circ $ → (rad)
$ 15 \times \frac{\pi}{180} = \frac{\pi}{12} $ rad
✔ Answer: $ \frac{\pi}{12} $ rad

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e) $ 12^\circ $ → (°)
Already in degrees.
✔ Answer: $ 12^\circ $

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f) $ 324^\circ $ → (rad)
$ 324 \times \frac{\pi}{180} = \frac{324\pi}{180} = \frac{9\pi}{5} $ rad
✔ Answer: $ \frac{9\pi}{5} $ rad

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g) $ 22^\circ 30' $ → (rad)
Convert minutes to decimal degrees:
$ 30' = \frac{30}{60} = 0.5^\circ $
So total: $ 22.5^\circ $
Now to radians:
$ 22.5 \times \frac{\pi}{180} = \frac{22.5\pi}{180} = \frac{\pi}{8} $ rad
✔ Answer: $ \frac{\pi}{8} $ rad

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h) $ 0.5' $ → (')
Already in minutes.
✔ Answer: $ 0.5' $

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i) $ 2.005\pi $ → (°')
Convert to degrees:
$ 2.005\pi \times \frac{180}{\pi} = 2.005 \times 180 = 360.9^\circ $
Now break into degrees and minutes:
$ 0.9^\circ = 0.9 \times 60 = 54' $
So: $ 360^\circ 54' $
✔ Answer: $ 360^\circ 54' $

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j) $ 12.5\pi $ → (°)
$ 12.5\pi \times \frac{180}{\pi} = 12.5 \times 180 = 2250^\circ $
✔ Answer: $ 2250^\circ $

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k) $ 3.22\pi $ → (°')
Convert to degrees:
$ 3.22\pi \times \frac{180}{\pi} = 3.22 \times 180 = 579.6^\circ $
Now split:
$ 0.6^\circ = 0.6 \times 60 = 36' $
So: $ 579^\circ 36' $
✔ Answer: $ 579^\circ 36' $

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l) $ 292^\circ 30' $ → (rad)
Convert to decimal degrees:
$ 30' = 0.5^\circ $ → $ 292.5^\circ $
To radians:
$ 292.5 \times \frac{\pi}{180} = \frac{292.5\pi}{180} = \frac{117\pi}{72} = \frac{13\pi}{8} $ rad
✔ Answer: $ \frac{13\pi}{8} $ rad

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m) $ 0.8\pi $ → (°)
$ 0.8\pi \times \frac{180}{\pi} = 0.8 \times 180 = 144^\circ $
✔ Answer: $ 144^\circ $

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n) $ 9,450' $ → (rad)
Convert to degrees:
$ 9,450' = \frac{9450}{60} = 157.5^\circ $
To radians:
$ 157.5 \times \frac{\pi}{180} = \frac{157.5\pi}{180} = \frac{7\pi}{8} $ rad
✔ Answer: $ \frac{7\pi}{8} $ rad

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o) $ 3.6\pi $ → (')
Convert to degrees:
$ 3.6\pi \times \frac{180}{\pi} = 3.6 \times 180 = 648^\circ $
Now to minutes:
$ 648^\circ \times 60 = 38,880' $
✔ Answer: $ 38,880' $

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p) $ 7.29\pi $ → (°')
Convert to degrees:
$ 7.29\pi \times \frac{180}{\pi} = 7.29 \times 180 = 1312.2^\circ $
Break down:
$ 0.2^\circ = 0.2 \times 60 = 12' $
So: $ 1312^\circ 12' $
✔ Answer: $ 1312^\circ 12' $

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q) $ 1,000' $ → (rad)
Convert to degrees:
$ 1000' = \frac{1000}{60} \approx 16.6667^\circ $
To radians:
$ 16.6667 \times \frac{\pi}{180} \approx \frac{16.6667\pi}{180} = \frac{5\pi}{54} $ rad
(Exact: $ \frac{1000}{60} = \frac{50}{3} $, so $ \frac{50}{3} \times \frac{\pi}{180} = \frac{50\pi}{540} = \frac{\pi}{10.8} $, but better as $ \frac{5\pi}{54} $)
✔ Answer: $ \frac{5\pi}{54} $ rad

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r) $ 6.23^\circ $ → (°)
Already in degrees.
✔ Answer: $ 6.23^\circ $

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s) $ 23.1\pi $ → (°)
$ 23.1\pi \times \frac{180}{\pi} = 23.1 \times 180 = 4158^\circ $
✔ Answer: $ 4158^\circ $

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t) $ 790^\circ $ → (rad)
$ 790 \times \frac{\pi}{180} = \frac{790\pi}{180} = \frac{79\pi}{18} $ rad
✔ Answer: $ \frac{79\pi}{18} $ rad

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u) $ 17^\circ $ → (°)
Already in degrees.
✔ Answer: $ 17^\circ $

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v) $ 6,480' $ → (rad)
Convert to degrees:
$ 6,480' = \frac{6480}{60} = 108^\circ $
To radians:
$ 108 \times \frac{\pi}{180} = \frac{3\pi}{5} $ rad
✔ Answer: $ \frac{3\pi}{5} $ rad

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w) $ 5.25\pi $ → (°)
$ 5.25\pi \times \frac{180}{\pi} = 5.25 \times 180 = 945^\circ $
✔ Answer: $ 945^\circ $

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x) $ 0.1^\circ $ → (rad)
$ 0.1 \times \frac{\pi}{180} = \frac{\pi}{1800} $ rad
✔ Answer: $ \frac{\pi}{1800} $ rad

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y) $ 25.12^\circ $ → (°)
Already in degrees.
✔ Answer: $ 25.12^\circ $

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z) $ 101^\circ 15' $ → (rad)
Convert to decimal degrees:
$ 15' = \frac{15}{60} = 0.25^\circ $ → $ 101.25^\circ $
To radians:
$ 101.25 \times \frac{\pi}{180} = \frac{101.25\pi}{180} = \frac{405\pi}{720} = \frac{9\pi}{16} $ rad
✔ Answer: $ \frac{9\pi}{16} $ rad

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Z) $ 0.015' $ → (')
Already in minutes.
✔ Answer: $ 0.015' $

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Final Answers Summary:

| Item | Answer |
|------|--------|
| a) | $ \pi $ rad |
| b) | $ 540^\circ $ |
| c) | $ \frac{2\pi}{3} $ rad |
| d) | $ \frac{\pi}{12} $ rad |
| e) | $ 12^\circ $ |
| f) | $ \frac{9\pi}{5} $ rad |
| g) | $ \frac{\pi}{8} $ rad |
| h) | $ 0.5' $ |
| i) | $ 360^\circ 54' $ |
| j) | $ 2250^\circ $ |
| k) | $ 579^\circ 36' $ |
| l) | $ \frac{13\pi}{8} $ rad |
| m) | $ 144^\circ $ |
| n) | $ \frac{7\pi}{8} $ rad |
| o) | $ 38,880' $ |
| p) | $ 1312^\circ 12' $ |
| q) | $ \frac{5\pi}{54} $ rad |
| r) | $ 6.23^\circ $ |
| s) | $ 4158^\circ $ |
| t) | $ \frac{79\pi}{18} $ rad |
| u) | $ 17^\circ $ |
| v) | $ \frac{3\pi}{5} $ rad |
| w) | $ 945^\circ $ |
| x) | $ \frac{\pi}{1800} $ rad |
| y) | $ 25.12^\circ $ |
| z) | $ \frac{9\pi}{16} $ rad |
| Z) | $ 0.015' $ |

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