The image contains problems asking for conversions from radians to degrees and determining the "principal value" of given angles in radians.

Here’s a step-by-step breakdown of what each part represents:

Converting from Radians to Degrees:

To convert radians to degrees, use the formula: $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

For each question (a-g) in the left section:

1. a) $\frac{\pi}{8} \, \text{radians}$ $\text{Degrees} = \frac{\pi}{8} \times \frac{180}{\pi} = \frac{180}{8} = 22.5^\circ$

2. b) $\frac{3\pi}{2} \, \text{radians}$ $\text{Degrees} = \frac{3\pi}{2} \times \frac{180}{\pi} = 270^\circ$

3. c) $\frac{4\pi}{5} \, \text{radians}$ $\text{Degrees} = \frac{4\pi}{5} \times \frac{180}{\pi} = 144^\circ$

4. d) $\frac{\pi}{5} \, \text{radians}$ $\text{Degrees} = \frac{\pi}{5} \times \frac{180}{\pi} = 36^\circ$

5. e) $\frac{7\pi}{2} \, \text{radians}$ $\text{Degrees} = \frac{7\pi}{2} \times \frac{180}{\pi} = 630^\circ$

6. f) $\frac{12\pi}{5} \, \text{radians}$ $\text{Degrees} = \frac{12\pi}{5} \times \frac{180}{\pi} = 432^\circ$

7. g) $\frac{12\pi}{5} \, \text{radians}$ (Same as f, so it also equals 432°)

Principal Values (Esas Ölçü):

The "principal value" or "esas ölçü" refers to reducing an angle to the range $0^\circ \leq \theta < 360^\circ$.

For each question in the right section (A-F):

1. A) $\frac{7\pi}{6} \, \text{radians}$ $\text{Degrees} = \frac{7\pi}{6} \times \frac{180}{\pi} = 210^\circ$ Principal value: $210^\circ$ (since it's already within the 0° to 360° range).

2. B) $\frac{7\pi}{6} \, \text{radians}$ (same as A) Principal value: $210^\circ$

3. C) $\frac{38\pi}{5} \, \text{radians}$ $\text{Degrees} = \frac{38\pi}{5} \times \frac{180}{\pi} = 1368^\circ$ Now, subtract multiples of 360°: $1368^\circ - 3 \times 360^\circ = 288^\circ$ Principal value: $288^\circ$

4. D) $\frac{38\pi}{5} \, \text{radians}$ (same as C) Principal value: $288^\circ$

5. E) $\frac{9\pi}{2} \, \text{radians}$ $\text{Degrees} = \frac{9\pi}{2} \times \frac{180}{\pi} = 810^\circ$ Subtracting multiples of 360°: $810^\circ - 2 \times 360^\circ = 90^\circ$ Principal value: $90^\circ$

6. F) $\frac{21\pi}{2} \, \text{radians}$ $\text{Degrees} = \frac{21\pi}{2} \times \frac{180}{\pi} = 1890^\circ$ Subtracting multiples of 360°: $1890^\circ - 5 \times 360^\circ = 90^\circ$ Principal value: $90^\circ$

Summary of Principal Values:

• A) 210°
• B) 210°
• C) 288°
• D) 288°
• E) 90°
• F) 90°

Let me know if you would like further clarifications on any specific part of the problem.

Here are 5 related questions for more practice:

1. Convert $\frac{5\pi}{4} \, \text{radians}$ to degrees.
2. What is the principal value of $5\pi \, \text{radians}?$
3. How many degrees are there in $\frac{2\pi}{3} \, \text{radians}?$
4. What is the principal value of $\frac{15\pi}{4} \, \text{radians}?$
5. Convert $\frac{8\pi}{6} \, \text{radians}$ to degrees and find its principal value.

Tip:

Always reduce radians by subtracting multiples of $2\pi$ (or $360^\circ$ for degrees) to find the principal value of an angle.