Converting Radians To Degrees and Vice Versa | PDF | Trigonometric ... - Free Printable

You've uploaded a worksheet with repeated sections — the same conversion problems appear multiple times. I’ll solve all unique problems once, clearly explaining the method, and then provide the full answer key for easy reference.

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## 🔄 Conversion Formulas

Degrees to Radians:

\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\]

Radians to Degrees:

\[
\text{degrees} = \text{radians} \times \frac{180}{\pi}
\]

We’ll leave radian answers as multiples of π (e.g., \(\frac{\pi}{3}\)), and degree answers as numbers (e.g., 60°).

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## ✔ PART 1: Convert Degrees to Radians (Problems 1–16)

| Degree | Radian (as multiple of π) | Calculation |
|--------|---------------------------|-----------|
| 1. 60° | \(\frac{\pi}{3}\) | \(60 \times \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3}\) |
| 2. 30° | \(\frac{\pi}{6}\) | \(30 \times \frac{\pi}{180} = \frac{\pi}{6}\) |
| 3. 90° | \(\frac{\pi}{2}\) | \(90 \times \frac{\pi}{180} = \frac{\pi}{2}\) |
| 4. 120°| \(\frac{2\pi}{3}\) | \(120 \times \frac{\pi}{180} = \frac{2\pi}{3}\) |
| 5. 150°| \(\frac{5\pi}{6}\) | \(150 \times \frac{\pi}{180} = \frac{5\pi}{6}\) |
| 6. 135°| \(\frac{3\pi}{4}\) | \(135 \times \frac{\pi}{180} = \frac{3\pi}{4}\) |
| 7. 210°| \(\frac{7\pi}{6}\) | \(210 \times \frac{\pi}{180} = \frac{7\pi}{6}\) |
| 8. 270°| \(\frac{3\pi}{2}\) | \(270 \times \frac{\pi}{180} = \frac{3\pi}{2}\) |
| 9. 300°| \(\frac{5\pi}{3}\) | \(300 \times \frac{\pi}{180} = \frac{5\pi}{3}\) |
| 10. 315°| \(\frac{7\pi}{4}\) | \(315 \times \frac{\pi}{180} = \frac{7\pi}{4}\) |
| 11. 450°| \(\frac{5\pi}{2}\) | \(450 \times \frac{\pi}{180} = \frac{5\pi}{2}\) |
| 12. 480°| \(\frac{8\pi}{3}\) | \(480 \times \frac{\pi}{180} = \frac{8\pi}{3}\) |
| 13. 20° | \(\frac{\pi}{9}\) | \(20 \times \frac{\pi}{180} = \frac{\pi}{9}\) |
| 14. 80° | \(\frac{4\pi}{9}\) | \(80 \times \frac{\pi}{180} = \frac{4\pi}{9}\) |
| 15. 140°| \(\frac{7\pi}{9}\) | \(140 \times \frac{\pi}{180} = \frac{7\pi}{9}\) |
| 16. 320°| \(\frac{16\pi}{9}\) | \(320 \times \frac{\pi}{180} = \frac{16\pi}{9}\) |

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## ✔ PART 2: Convert Radians to Degrees (Problems 1–15)

| Radian | Degree | Calculation |
|--------|--------|-----------|
| 1. \(\frac{\pi}{3}\) | 60° | \(\frac{\pi}{3} \times \frac{180}{\pi} = 60\) |
| 2. \(\frac{8\pi}{3}\) | 480° | \(\frac{8\pi}{3} \times \frac{180}{\pi} = 480\) |
| 3. \(\frac{7\pi}{4}\) | 315° | \(\frac{7\pi}{4} \times \frac{180}{\pi} = 315\) |
| 4. \(\frac{2\pi}{3}\) | 120° | \(\frac{2\pi}{3} \times \frac{180}{\pi} = 120\) |
| 5. \(\frac{11\pi}{6}\) | 330° | \(\frac{11\pi}{6} \times \frac{180}{\pi} = 330\) |
| 6. \(\frac{15\pi}{4}\) | 675° | \(\frac{15\pi}{4} \times \frac{180}{\pi} = 675\) |
| 7. \(-\frac{\pi}{6}\) | -30° | \(-\frac{\pi}{6} \times \frac{180}{\pi} = -30\) |
| 8. \(-\frac{\pi}{4}\) | -45° | \(-\frac{\pi}{4} \times \frac{180}{\pi} = -45\) |
| 9. \(\frac{8\pi}{5}\) | 288° | \(\frac{8\pi}{5} \times \frac{180}{\pi} = 288\) |
| 10. \(\frac{7\pi}{10}\) | 126° | \(\frac{7\pi}{10} \times \frac{180}{\pi} = 126\) |
| 11. \(\frac{11\pi}{15}\) | 132° | \(\frac{11\pi}{15} \times \frac{180}{\pi} = 132\) |
| 12. \(\frac{4\pi}{15}\) | 48° | \(\frac{4\pi}{15} \times \frac{180}{\pi} = 48\) |
| 13. \(\frac{7\pi}{20}\) | 63° | \(\frac{7\pi}{20} \times \frac{180}{\pi} = 63\) |
| 14. \(\frac{17\pi}{20}\) | 153° | \(\frac{17\pi}{20} \times \frac{180}{\pi} = 153\) |
| 15. \(\frac{11\pi}{30}\) | 66° | \(\frac{11\pi}{30} \times \frac{180}{\pi} = 66\) |

> Note: Problem 15 in the first section is labeled “15π/32” — that’s likely a typo or misnumbering. Since it doesn’t match any other problem, and assuming it was meant to be #15 in the radian-to-degree section, I’ve included it here as #15 above. If you meant to include \(\frac{15\pi}{32}\), here’s its value:

- \(\frac{15\pi}{32} \times \frac{180}{\pi} = \frac{15 \times 180}{32} = \frac{2700}{32} = 84.375^\circ\)

But since the rest are whole numbers, this may be an error. I’ll assume it was meant to be \(\frac{11\pi}{30}\) as listed.

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## 🔁 REPEATED SECTIONS

The worksheet repeats the same problems under different headings (e.g., Problems 17–32 are identical to 1–16; 33–48 same again). So we don’t need to re-solve them — just reuse the answers above.

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## 📋 FINAL ANSWER KEY (All Unique Problems)

Degrees → Radians (Problems 1–16):

1. \(\frac{\pi}{3}\)
2. \(\frac{\pi}{6}\)
3. \(\frac{\pi}{2}\)
4. \(\frac{2\pi}{3}\)
5. \(\frac{5\pi}{6}\)
6. \(\frac{3\pi}{4}\)
7. \(\frac{7\pi}{6}\)
8. \(\frac{3\pi}{2}\)
9. \(\frac{5\pi}{3}\)
10. \(\frac{7\pi}{4}\)
11. \(\frac{5\pi}{2}\)
12. \(\frac{8\pi}{3}\)
13. \(\frac{\pi}{9}\)
14. \(\frac{4\pi}{9}\)
15. \(\frac{7\pi}{9}\)
16. \(\frac{16\pi}{9}\)

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Radians → Degrees (Problems 1–15):

1. 60°
2. 480°
3. 315°
4. 120°
5. 330°
6. 675°
7. -30°
8. -45°
9. 288°
10. 126°
11. 132°
12. 48°
13. 63°
14. 153°
15. 66°

*(If you meant \(\frac{15\pi}{32}\) instead of #15, answer is 84.375°)*

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✔ All problems solved! Since the worksheet repeats, simply copy these answers into each corresponding section.

Let me know if you’d like a printable version or step-by-step work for each problem!