Angles

1. 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.

2. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

3.

4.

5.

6.

7.

8. 240°

9. [latex]\frac{4\pi }{3}[/latex]

10. [latex]\frac{2\pi }{3}[/latex]

11. [latex]s=\frac{7\pi}{3}\approx 7.33[/latex]

12. [latex]\frac{7\pi }{2}\approx 11.00{\text{ in}}^{2}[/latex]

13. [latex]s=\frac{9\pi}{5}\approx 5.65[/latex]

14. [latex]\frac{81\pi }{20}\approx 12.72{\text{ cm}}^{2}[/latex]

15. 20°

16. 60°

17. −75°

18. [latex]\frac{\pi }{2}[/latex] radians

19. [latex]-3\pi[/latex] radians

[unmatched: no corresponding problem shown between #35 and #39 in the exercise list — key answer is “π radians”]

20. [latex]\frac{5\pi }{6}[/latex] radians

21. [latex]\frac{5.02\pi }{3}\approx 5.26[/latex] miles

22. [latex]\frac{21\pi }{10}\approx 6.60[/latex] meters

23. 104.7198 cm2

24. 0.7697 in2

25. 250°

26. 320°

27. [latex]\frac{4\pi }{3}[/latex]

28. [latex]\frac{8\pi }{9}[/latex]

29. 1320 rad 210.085 RPM

30. 7 in./s, 4.77 RPM, 28.65 deg/s

31. [latex]5.76[/latex] miles

32. [latex]120^\circ[/latex]

33. 11.5 inches

34. [latex]52^\circ 22' 30''[/latex]

35. [latex]-73^\circ 15' 0''[/latex]

36. [latex]140^\circ 0' 28.8''[/latex]

37. [latex]38.2567^\circ[/latex]

38. [latex]72.0100^\circ[/latex]

39. [latex]-19.5000^\circ[/latex]

40. Complement: [latex]62^\circ[/latex], Supplement: [latex]152^\circ[/latex]

41. Complement: [latex]47^\circ 41' 24''[/latex], Supplement: [latex]137^\circ 41' 24''[/latex]

42. Complement: No complement exists, Supplement: [latex]68^\circ[/latex]

43. Complement: [latex]\frac{3\pi}{8}[/latex], Supplement: [latex]\frac{7\pi}{8}[/latex]

Unit Circle: Sine and Cosine Functions

1. On the unit circle, the coordinates of a point corresponding to an angle t are [latex](\cos t,\ \sin t)[/latex]. The x-coordinate represents [latex]\cos t[/latex] and the y-coordinate represents [latex]\sin t[/latex].

2. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, [latex]t[/latex], formed by the terminal side of the angle [latex]t[/latex] and the horizontal axis.

3. The sine values are equal.

4. I

5. IV

6. [latex]\frac{\sqrt{3}}{2}[/latex]

7. [latex]\frac{1}{2}[/latex]

8. [latex]\frac{\sqrt{2}}{2}[/latex]

9. 0

10. [latex]\frac{\sqrt{3}}{2}[/latex]

11. [latex]60^\circ[/latex]

12. [latex]45^\circ[/latex]

13. [latex]\frac{\pi }{3}[/latex]

14. [latex]\frac{\pi }{3}[/latex]

15. [latex]60^\circ[/latex], Quadrant IV, [latex]\text{sin}\left(300^\circ \right)=-\frac{\sqrt{3}}{2},\cos \left(300^\circ \right)=\frac{1}{2}[/latex]

16. [latex]45^\circ[/latex], Quadrant II, [latex]\text{sin}\left(135^\circ \right)=\frac{\sqrt{2}}{2}[/latex], [latex]\cos \left(135^\circ \right)=-\frac{\sqrt{2}}{2}[/latex]

17. [latex]\frac{\pi }{6}[/latex], Quadrant III, [latex]\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2}[/latex], [latex]\text{cos}\left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}[/latex]

18. [latex]\frac{\pi }{4}[/latex], Quadrant II, [latex]\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}[/latex], [latex]\cos \left(\frac{4\pi }{3}\right)=-\frac{\sqrt[]{2}}{2}[/latex]

19. [latex]\frac{\sqrt{77}}{9}[/latex]

20. [latex]-\frac{\sqrt{15}}{4}[/latex]

21. [latex]\sin t=\frac{1}{2},\cos t=-\frac{\sqrt{3}}{2}[/latex]

22. [latex]\sin t=-\frac{\sqrt{2}}{2},\cos t=-\frac{\sqrt{2}}{2}[/latex]

23. [latex]\sin t=\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2}[/latex]

24. [latex]\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2}[/latex]

25. [latex]\sin t=0,\cos t=-1[/latex]

26. [latex]\sin t=\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}[/latex]

27. −0.1736

28. 0.9511

29. −0.1392

30. −0.7660

Other Trigonometric Functions

1. [latex]\frac{2\sqrt{3}}{3}[/latex]

2. 1

3. [latex]-\frac{2\sqrt{3}}{3}[/latex]

4. [latex]\sqrt{3}[/latex]

5. [latex]\frac{\sqrt{3}}{3}[/latex]

6. −2

7. If [latex]\sin t=-\frac{2\sqrt{2}}{3},\sec t=-3,\csc t=-\frac{3\sqrt{2}}{4},\tan t=2\sqrt{2},\cot t=\frac{\sqrt{2}}{4}[/latex]

8. [latex]\sec t=2,\csc t=\frac{2\sqrt{3}}{3},\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3}[/latex]

9. [latex]-\frac{\sqrt{2}}{2}[/latex]

10. 3.1

11. 1.4

12. –0.228

13. –2.414

14. 1.414

15. 1.556

16. 13.77 hours, period: [latex]1000\pi[/latex]

17. 7.73 inches