1. Write the circular (radian) measure of the angle produced by the tip of a clock’s hour hand after rotating for 1 hour.

2. If the sum of two angles is 135° and their difference is \(\cfrac{\pi}{12}\), then calculate and write the measures of the two angles in both sexagesimal (degree-minute-second) and radian systems.

3. If the circular (radian) measure of an angle is \( \frac{7\pi}{12} \), what is its value in the sexagesimal (degree) system?

(a) 90° (b) 105° (c) 135° (d) 160°

4. A rotating ray, starting from a certain position, rotates two full turns in the counterclockwise direction (opposite to the clock hands) and then produces an additional angle of 30°. What are the sexagesimal (degree) and circular (radian) measures of this angle?

5. The radius of a circle is 28 cm. Find the radian measure of the central angle subtended by an arc of length 5.5 cm.

6. If the radius of a circle is 7 cm, then find the radian measure of the central angle subtended by an arc of length 5.5 cm.

7. If the radius of a circle is 6 cm, calculate and write the radian measure of the central angle subtended at the center by an arc of length 15 cm.

8. In triangle \( \triangle ABC \), AC = BC and side BC is extended up to point D. If \( \angle ACD = 144^\circ \), then find the radian measure of each angle of triangle ABC.

9. In a triangle, one angle measures \(65^\circ\) and the second angle measures \(\cfrac{\pi}{12}\); calculate and write the measure of the third angle in both sexagesimal (degree-minute-second) and radian systems.

10. The radius of a circle is 28 cm. In this circle, calculate and write the radian measure of the central angle subtended by an arc of length 5.5 cm.

11. The tip of a clock’s minute hand rotates in 1 hour.

(a) \(\cfrac{π}{4}\) radians (b) \(\cfrac{π}{2}\)radians (c) \(π\) radians (d) \(2π\) radians

12. The measure of the central angle formed at the center of a circle by an arc whose length is equal to the radius of the circle is 1 radian.

13. A rotating ray turns in the counterclockwise direction, completing one full revolution followed by a half revolution and then produces a \(30^\circ\) angle. What is the radian measure of the angle in trigonometric terms?

14. The tip of the minute hand of a clock — how many radians does it rotate in one hour?

15. The difference in measurement between two angles is 1°, and the sum of their radian measures is 1 radian. Express the angles in degrees.

16. In a circle with center \(O\), \(AB\) is a diameter. \(P\) is any point on the circumference. If \(\angle POA = 120°\), then the measurement of \(\angle PBO\) is –

(a) 30° (b) 60° (c) 90° (d) 120°

17. After a spherical ice ball starts melting, it is observed that its curved surface area has reduced by 75% after 15 minutes. The percentage of ice that has melted during this time is –

(a) 75 (b) 87.5 (c) 50 (d) 92.5

18. If the measure of an angle in degrees is \(x\) and in radians is \(y\), then what is the value of \( \frac{x}{y} \)?

(a) \(\cfrac{π}{180}\) (b) \(\cfrac{π}{90}\) (c) \(\cfrac{90}{π}\) (d) \(\cfrac{180}{π}\)

19. In the cyclic quadrilateral PQRS, the side PS is a diameter of the circle. If \(\angle\)PQR = 120°, then what is the measure of \(\angle\)SPR?

(a) 90° (b) 30° (c) 60° (d) 120°

20. In triangle ABC, AB = AC. A line drawn from point C intersects the extended line BA at point D. If AC = AD, what is the measure of \(\angle\)BCD in radians?

(a) \(\cfrac{π}{2}\) (b) \(π\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{3}\)

21. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

22. In the cyclic quadrilateral PORS, the side PS is a diameter of the circle. If \(\angle\)PQR = 128°, then what is the measure of \(\angle\)SPR?

(a) 30° (b) 38° (c) 60° (d) None of the above

23. In triangle ABC, the incenter is O. If \(\angle\)BOC = 120°, then what is the measure of \(\angle\)BAC?

(a) 60° (b) 90° (c) 45° (d) 120°

24. What will be the measure of the third angle of a triangle in radians if the other two angles are \(65^\circ 56' 44''\) and \(64^\circ 3' 16''\)?

25. If the sum of a central angle and an inscribed angle of a circle is 180°, then what is the measure of the central angle?

26. In an equilateral triangle ABC, the base BC is extended to a point E such that CE = BC. A is joined to E to form triangle ACE. Find the circular (radian) measures of the angles of triangle ACE.

27. Two unequal arcs of a circle subtend two angles at the center in the ratio 5:3, and the sexagesimal (degree) measure of the second angle is 45°. What is the radian measure of the first angle?

28. The radian measure of the supplementary angle of an angle measuring \(\cfrac{3π}{8}\) is ____.

29. The radius of a circle is 28 cm. What is the radian measure of the central angle subtended by an arc of length 5.5 cm on this circle?

30. If the sum of two angles is 135° and their difference is \(\frac{π}{12}\), write the angles in sexagesimal and circular measure.