5.1e: Exercises - Angles

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A: Concepts

Exercise $\PageIndex{A}$

1) Draw an angle in standard position. Label the vertex, initial side, and terminal side.

2) Explain why there are an infinite number of angles that are coterminal to a certain angle.

3) State what a positive or negative angle signifies, and explain how to draw each.

4) How does radian measure of an angle compare to the degree measure? Include an explanation of $1$ radian in your paragraph.

Answers to odd exercises:: 1. $Graph of a circle with an angle inscribed, showing the initial side, terminal side, and vertex.$
3. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

B: Draw Angles in Standard Position

Exercise $\PageIndex{B}$

$ \bigstar $ Draw an angle in standard position with the given measure. If the angle measure is not between $0$ and $2 \pi $ or between $0^{\circ}$ and $360^{\circ}$, also state the coterminal angle that is within that interval.

5. $300^{\circ}$

6. $415^{\circ}$

7. $135^{\circ}$

8. $-30^{\circ}$

9. $-120^{\circ}$

10. $-315^{\circ}$

11. $\dfrac{2π}{3}$

12. $\dfrac{5π}{3}$

13. $−\dfrac{4π}{3}$

14. $−\dfrac{5π}{3}$

15. $\dfrac{5π}{6}$

16. $\dfrac{11π}{6}$

17. $−\dfrac{π}{6}$

18. $\dfrac{7π}{6}$

19. $ \dfrac{7π}{4}$

20. $\dfrac{5π}{4}$

21. $−\dfrac{3π}{4}$

22. $\dfrac{11π}{4}$

23. $\dfrac{22π}{3}$

24. $ \dfrac{17π}{6} $

25. $−\dfrac{π}{10}$

26. $\dfrac{23π}{5}$

27. $\dfrac{π}{2}$

28. $ -\dfrac{3π}{2}$

29. $ \dfrac{7π}{2}$

30. $ -5π $

Answers to odd exercises:

5. $Graph of a circle with an angle inscribed.$	7. $Graph of a circle with a 135 degree angle inscribed.$	9. $240^{\circ}$ $Graph of a circle showing the equivalence of two angles.$	11. $Graph of a circle with a 2pi/3 radians angle inscribed.$	13. $\frac{2π}{3}$ $Graph of a circle showing the equivalence of two angles.$ x
15. $Graph of a circle with 5pi/6 radians angle inscribed.$	17. $5.1e 17.png$	19. $5.1 19.png$	21. $5.1e 21.png$
23. $\frac{4π}{3}$ $Graph of a circle showing the equivalence of two angles.$	25. $\frac{19π}{10}$ $Graph of a circle with a –pi/10 radians angle inscribed.$	27. $5.1e 27.png$	29 . $\frac{3π}{2}$ $5.1e 29.png$

C: Convert between radians and degrees

Exercise $\PageIndex{C}$

$ \bigstar $ Convert angles in radians to degrees.

36. $\dfrac{3π}{4}$ radians

37. $\dfrac{π}{9}$ radians

38. $−\dfrac{5π}{4}$ radians

39. $\dfrac{π}{3}$ radians

40. $−\dfrac{7π}{3}$ radians

41. $−\dfrac{5π}{12}$ radians

42. $\dfrac{11π}{6}$ radians

$ \bigstar $ Convert angles in degrees to radians. Write the answer both as a multiple of pi and to the nearest hundredth of a radian.

43. $90^{\circ}$

44. $100^{\circ}$

45. $-540^{\circ}$

46. $-120^{\circ}$

47. $180^{\circ}$

48. $-315^{\circ}$

49. $150^{\circ}$

Answers to odd exercises:: 37. $20^{\circ} \qquad $ 39. $60^{\circ} \qquad $ 41. $-75^{\circ} \qquad $ 43. $\frac{π}{2} \; \approx \; 1.57$ radians $ \qquad $ 45. $−3π \; \approx \; 9.42$ radians
47. $π \; \approx \; 3.14 $ radians $ \qquad $ 49. $\frac{5π}{6} \; \approx \; 2.62$ radians

D: Coterminal Angles

Exercise $\PageIndex{D}$

$ \bigstar $ Find the angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal to the given angle.

50. $-40^{\circ}$ $ \qquad $ 51. $-110^{\circ}$ $ \qquad $ 52. $700^{\circ}$ $ \qquad $ 53. $1400^{\circ}$

$ \bigstar $ Find the angle between $0$ and $2\pi $ in radians that is coterminal to the given angle.

54. $−\dfrac{π}{9}$ $ \qquad $ 55. $\dfrac{10π}{3}$ $ \qquad $ 56. $\dfrac{13π}{6}$ $ \qquad $ 57. $\dfrac{44π}{9}$

Answers to odd exercises:: 51. $250^{\circ}$ $ \qquad $ 53. $320^{\circ}$ $ \qquad $ 55. $\frac{4π}{3}$ $ \qquad $ 57. $\frac{8π}{9}$

5. $Graph of a circle with an angle inscribed.$	7. $Graph of a circle with a 135 degree angle inscribed.$	9. \(240^{\circ}\) $Graph of a circle showing the equivalence of two angles.$	11. $Graph of a circle with a 2pi/3 radians angle inscribed.$	13. \(\frac{2π}{3}\) $Graph of a circle showing the equivalence of two angles.$ x
15. $Graph of a circle with 5pi/6 radians angle inscribed.$	17. $5.1e 17.png$	19. $5.1 19.png$	21. $5.1e 21.png$
23. \(\frac{4π}{3}\) $Graph of a circle showing the equivalence of two angles.$	25. \(\frac{19π}{10}\) $Graph of a circle with a –pi/10 radians angle inscribed.$	27. $5.1e 27.png$	29 . \(\frac{3π}{2}\) $5.1e 29.png$