Trigonometry Final Review Exercises

Transcription

1 1 The exam will last 2 hours and will be entirely short answer. Be sure to provide adequate written work to justify your answers in order to receive full credit. You will be provided with a basic trig identity sheet. 1. Draw the angles and -2 in standard position. Label both. [NOTICE: there s no degree mark on -2, so it is assumed to be in radians] 2. How long is the arc that subtends a central angle of in a circle of radius 5 inches? Give an exact answer and a decimal rounded to the nearest hundredth. 3. What is the area of the sector whose central angle is in a circle of radius 5 inches? Give an exact answer and a decimal rounded to the nearest hundredth. 4. The windshield wiper of a car is 24 inches long. The area that the windshield wiper wipes clean is 675 square inches. Find the measure of the central angle swept out by the windshield wiper in degrees. Round to the nearest degree. 5. A merry-go-round makes three complete circles each minute. Find the angular velocity of a child standing 8 feet from the center of the merry-go-round. 6. A neighborhood carnival has a Ferris wheel whose radius is 32 feet. You measure the time it takes for one revolution to be 75 seconds. What is the linear speed of a point on the rim of the Ferris wheel in feet per second?

2 2 7. Given a right triangle with legs of length and and a hypotenuse of length, and an angle as in the diagram below, define the six trigonometric functions in terms of,, and by filling in the blanks: 8. Consider an angle in standard position. If and is in Quadrant III, find the other trig function values. (Fill in the blank.) 2/9 9. Solve the right triangle where, and. Round to the nearest tenth of a degree and foot ft The angle of depression from the top of a building to a point on the ground is. A foot cable reaches from the top of the building to that point. How far is the point on the ground from the base of the building?

3 11. Complete the following table. Trigonometry Final Review Exercises 3 Function Domain Range ( ) 12. Draw the graph of ( ) for at least one period. Label key points or make sure that the coordinates are clearly identifiable by the labels on your axes. 13. Draw the graph of ( ) for at least one period. Label key points or make sure that the coordinates are clearly identifiable by the labels on your axes. 14. Draw the graph of ( ) for at least one period. Label key points or make sure that the coordinates are clearly identifiable by the labels on your axes.

4 15. Try the last three problems with all the different trig functions! Show that ( ) is not an identity. 17. Show that is an identity. 18. Compute the exact value of. (NO DECIMALS!) 19. Suppose that an angle is in Quadrant II with and an angle is in Quadrant III with. Compute the exact value of ( ). (NO DECIMALS!) 20. Prove the following identity or give a counterexample. 21. Prove the following identity or give a counterexample. ( )

5 5 22. Given that ( ), which are the possible values of ( )? 23. In what quadrant would you find ( )? 24. Complete the following table. Function Domain Range 25. Give the exact value of ( ( )). 26. Solve. 27. Solve for values in ) rounded to the nearest degree. 28. Solve.

6 29. Solve for angles in ). If appropriate, round to the nearest degree Solve the triangle(s), if any, that satisfy the given information:,,. Round final answers to the nearest tenth. [make sure to find both solutions] 31. Solve the triangle(s), if any, that satisfy the given information:,,. Round final answers that are angles to the nearest degree and final answers that are side lengths to the nearest tenth. 32. Solve the triangle(s), if any, that satisfy the given information: a,, c Round final answers that are angles to the nearest degree and final answers that are side lengths to the nearest tenth.

7 7 Trigonometry Final Review Exercises 33. Solve the triangle(s), if any, that satisfy the given information:,,. Round final answers that are angles to the nearest tenth of a degree and sides to the nearest tenth of a mile. 34. Find the area of the triangle for which,, and. Round to the nearest tenth of a square inch. 35. Find the area of the triangle for which,,. Round to the nearest tenth of a square inch. 36. Given vectors and, compute the following: a. b. c. ( ) 37. Compute the angle between the vectors and. Round to the nearest degree.

8 8 38. Perform the complex number operation and give your final answer as an exact complex number in standard form: ( )( ). 39. Perform the complex number operation and give your final answer as an exact complex number in standard form: ( ) 40. Write in trigonometric form. Give an approximate answer with r rounded to the nearest tenth and rounded to the nearest degree. 41. Multiply:. 42. Compute the three cube roots of. 43. Plot the polar point ( ).

9 44. Plot the polar point ( ) Graph the polar equation 46. Graph the polar equation. 47. Convert the rectangular point (15,-14) to polar. Round r to the nearest tenth and to the nearest degree. 48. Convert the polar point ( ) to rectangular. Give exact coordinates. 49. Convert the following polar equation to rectangular:. 50. Give the coordinates of five points on the graph of {.