392 CHAPTER 6 Trigonometric Functions Historical Feature Trigonometry was developed by Greek astronomers, who regarded the sky as the inside of a sphere, so it was natural that triangles on a sphere were investigated early (by Menelaus of Alexandria about AD 100) and that triangles in the plane were studied much later. The first book containing a systematic treatment of plane and spherical trigonometry was written by the Persian astronomer Nasir Eddin (about AD 1250). Regiomontanus (1436–1476) is the person most responsible for moving trigonometry from astronomy into mathematics. His work was improved by Copernicus (1473–1543) and Copernicus’s student Rhaeticus (1514–1576). Rhaeticus’s book was the first to define the six trigonometric functions as ratios of sides of triangles, although he did not give the functions their present names. Credit for this is due to Thomas Finck (1583), but Finck’s notation was by no means universally accepted at the time. The notation was finally stabilized by the textbooks of Leonhard Euler (1707–1783). Trigonometry has since evolved from its use by surveyors, navigators, and engineers to present applications involving ocean tides, the rise and fall of food supplies in certain ecologies, brain wave patterns, and many other phenomena. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 6.1 Assess Your Understanding 1. What is the formula for the circumference C of a circle of radius r ? What is the formula for the area A of a circle of radius r ? (p. A16) 2. If an object has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d = . (pp. A70–A71) Concepts and Vocabulary 3. An angle θ is in if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x -axis. 4. A is a positive angle whose vertex is at the center of a circle. 5. Multiple Choice If the radius of a circle is r and the length of the arc subtended by a central angle is also r, then the measure of the angle is 1 . (a) degree (b) minute (c) second (d) radian 6. On a circle of radius r, a central angle of θ radians subtends an arc of length s = ; the area of the sector formed by this angle θ is A = . 7. Multiple Choice 180° = radians (a) 2 π (b) π (c) 3 2 π (d) 2π 8. An object travels on a circle of radius r with constant speed. If s is the distance traveled in time t on the circle and θ is the central angle (in radians) swept out in time t, then the linear speed of the object is v = and the angular speed of the object is ω = . 9. True or False The angular speed ω of an object traveling on a circle of radius r is the angle θ (measured in radians) swept out, divided by the elapsed time t. 10. True or False For circular motion on a circle of radius r, linear speed equals angular speed divided by r. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Skill Building In Problems 11–22, draw each angle in standard position. 11. 30° 12. 60° 13. 135° 14. 120 − ° 15. 450° 16. 540° 17. 3 4 π 18. 4 3 π 19. 6 π − 20. 2 3 π − 21. 16 3 π 22. 21 4 π In Problems 23–34, convert each angle in degrees to radians. Express your answer as a multiple of .π 23. 30° 24. 120° 25. 495° 26. 330° 27. 60 − ° 28. 30 − ° 29. 540° 30. 270° 31. 240 − ° 32. 225 − ° 33. 90 − ° 34. 180 − ° In Problems 35–46, convert each angle in radians to degrees. 35. 3 π 36. 5 6 π 37. 13 6 π − 38. 2 3 π − 39. 9 2 π 40. 4π 41. 3 20 π 42. 5 12 π 43. 2 π − 44. π− 45. 17 15 π − 46. 3 4 π −
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