Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions (f) Give an interval, if one exists, on which f ( x ) = tan ( x ) is increasing. Then give an Answer the following. 1. (a) Use your calculator to complete the following chart. Round to the nearest hundredth. If a value is undefined, state \"Undefined.\" Notice that the x-values on the . chart increase by increments of 12 y = tan ( x ) x x y = tan ( x ) 0 2 7 12 12 interval , if one exists, on which f ( x ) = tan ( x ) is decreasing. 2. (a) Use your calculator to complete the following chart. Round to the nearest hundredth. If a value is undefined, state \"Undefined.\" Notice that the x-values on the . chart increase by increments of 12 6 5 6 12 5 12 11 12 6 7 12 2 3 3 4 4 (b) Plot the points from part (a) to discover the graph of f ( x ) = tan ( x ) . y = cot ( x ) 2 3 x 0 3 4 4 y = cot ( x ) x 2 3 3 5 6 5 12 11 12 4.0 y 3.0 2.0 1.0 x /12 /6 /4 /3 5/12 /2 7/12 2/3 3/4 5/6 11/12 (b) Plot the points from part (a) to discover the graph of g ( x ) = cot ( x ) . 1.0 4.0 y 2.0 3.0 3.0 2.0 4.0 1.0 x /12 (c) Use the graph from part (b) to sketch an extended graph of f ( x ) = tan ( x ) , where 2 x 2 . Be sure to show the intercepts as well as any asymptotes. (d) State the domain and range of f ( x ) = tan ( x ) . Do not base this answer simply on the limited domain from part (c), but the entire graph of f ( x ) = tan ( x ) . (e) State the period of f ( x ) = tan ( x ) . Math 1330, Precalculus The University of Houston /6 /4 /3 5/12 /2 7/12 2/3 3/4 5/6 11/12 1.0 2.0 3.0 4.0 (c) Use the graph from part (b) to sketch an extended graph of g ( x ) = cot ( x ) , where 2 x 2 . Be sure to show the intercepts as well as any asymptotes. Continued on the next page... Chapter 5: Trigonometric Functions Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions (d) State the domain and range of g ( x ) = cot ( x ) . Do not base this answer (c) Use the graph from part (b) to sketch an extended graph of f ( x ) = sec ( x ) , where 2 x 2 . Be sure to show the intercepts as well as any local maxima and minima. simply on the limited domain from part (c), but the entire graph of g ( x ) = cot ( x ) . (e) State the period of g ( x ) = cot ( x ) . (d) On the graph from part (c), superimpose the graph of h ( x ) = cos ( x ) lightly or in another (f) Give an interval, if one exists, on which g ( x ) = cot ( x ) is increasing. Then give an color. What do you notice? How can this serve as an aid in sketching the graph of f ( x ) = sec ( x ) ? interval , if one exists, on which g ( x ) = cot ( x ) is decreasing. 3. (e) State the domain and range of f ( x ) = sec ( x ) . Base this answer on the (a) Use your calculator to complete the following chart. Round to the nearest hundredth. If a value is undefined, state \"Undefined.\" y = sec ( x ) x x 0 7 6 6 entire graph of f ( x ) = sec ( x ) , not only on the partial graphs obtained in parts (b)-(d). y = sec ( x ) (f) State the period of f ( x ) = sec ( x ) . (g) Give two intervals on which f ( x ) = sec ( x ) is increasing. 5 4 4 4 3 3 2 3 2 2 3 5 3 3 4 7 4 5 6 11 6 4. (a) Use your calculator to complete the following chart. Round to the nearest hundredth. If a value is undefined, state \"Undefined.\" x 7 6 4 (b) Plot the points from part (a) to discover the graph of f ( x ) = sec ( x ) . 1.2 3 y 1.0 0.8 0.6 0.4 0.2 0.2 x /6 /3 /2 2/3 5/6 0.4 0.6 0.8 7/6 4/3 3/2 5/3 11/6 2 x 0 6 2 y = csc ( x ) y = csc ( x ) 5 4 4 3 2 3 2 2 3 5 3 3 4 7 4 5 6 11 6 2 1.0 1.2 Continued on the next page... Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions (b) Plot the points from part (a) to discover the graph of g ( x ) = csc ( x ) . 1.2 Matching. The left-hand column contains equations that represent transformations of f ( x ) = tan ( x ) . Match the equations on the left with the description on the right of how to obtain the graph of y = g ( x ) from y 1.0 the graph of f . 0.8 0.6 0.4 0.2 0.2 x /6 /3 /2 2/3 5/6 5. y = tan ( x 2 ) 6. y = tan ( x ) 2 7. y = tan ( 2 x ) 8. 1 y = tan ( x ) 2 9. y = tan ( 2 x 8 ) 7/6 4/3 3/2 5/3 11/6 2 0.4 0.6 0.8 1.0 1.2 A. Shrink horizontally by a factor of 12 . B. Reflect in the x-axis, then shift upward 2 units. C. Shift right 2 units. (c) Use the graph from part (b) to sketch an extended graph of g ( x ) = csc ( x ) , where 2 x 2 . Be sure to show the intercepts as well as any local maxima and minima. (d) On the graph from part (c), superimpose the graph of h ( x ) = sin ( x ) lightly or in another color. What do you notice? How can this serve as an aid in sketching the graph of g ( x ) = csc ( x ) ? (e) State the domain and range of g ( x ) = csc ( x ) . Base this answer on the entire graph of g ( x ) = csc ( x ) , not only on the partial graphs obtained in parts (b)-(d). (f) State the period of g ( x ) = csc ( x ) . 10. y = 2 tan ( x ) 1 11. y = tan x 2 x 12. y = tan 4 2 13. y = 2 tan ( 4 x 4 ) 14. y = tan ( x ) + 2 D. Reflect in the x-axis, stretch vertically by a factor of 2, shrink horizontally by a factor of 14 , then shift right 1 unit. E. Stretch horizontally by a factor of 2, then shift downward 4 units. F. Stretch horizontally by a factor of 2. G. Shrink horizontally by a factor of 12 , then shift right 4 units. H. Shift downward 2 units. I. Stretch vertically by a factor of 2. J. Shrink vertically by factor of 12 . (g) Give two intervals on which g ( x ) = csc ( x ) is increasing. Continued on the next page... Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions appropriate transformations on g ( x ) = tan ( x ) Matching. The left-hand column contains equations that represent transformations of f ( x ) = csc ( x ) . or h ( x ) = cot ( x ) within the following Match the equations on the left with the description on the right of how to obtain the graph of y = g ( x ) from intervals. (Note: The resulting graph may not fall within these intervals.) the graph of f . 15. y = 1 csc ( 3 x + ) 2 16. y = 1 csc 3 ( x + ) 2 17. y = 2 csc ( 3 x ) + 1 18. y = 2 csc x + 3 3 1 19. y = csc x + 3 3 1 20. y = csc x + 3 A. Reflect in the x-axis, stretch horizontally by a factor of 3, then shift left units. B. Shrink vertically by a factor of 12 , shrink horizontally by a factor of 1 3 , then shift left 3 units. C. Stretch vertically by a factor of 2, shrink horizontally by a factor of 13 , then shift upward units. D. Stretch vertically by a factor of 2, stretch horizontally by a factor of 3, then shift left 3 units. E. Reflect in the x-axis, stretch horizontally by a factor of 3, then shift upward units. F. Shrink vertically by a factor of 12 , shrink horizontally by a factor of 13 , then shift left units. (a) State the period. (b) Use transformations to sketch the graph of the function over one period. Label any asymptotes clearly. Be sure to show the exact transformation of each point on the basic graphs of g ( x ) = tan ( x ) or h ( x ) = cot ( x ) 4 2 < x< 2 h ( x ) = cot ( x ) , where 0 < x < 21. f ( x ) = 5 tan ( x ) 22. f ( x ) = 4 cot ( x ) 23. f ( x ) = 4 tan ( x ) + 1 24. f ( x ) = 5cot ( x ) 4 25. f ( x ) = 2 cot ( 3x ) 26. f ( x ) = 4 tan ( 2 x ) x 27. f ( x ) = 5cot + 2 4 x 28. f ( x ) = 3cot 5 2 29. f ( x ) = 7 tan ( 4 x ) 3 30. f ( x ) = 2 tan ( x ) + 1 31. f ( x ) = cot x + 3 32. f ( x ) = tan x 2 x 33. f ( x ) = 2 tan 5 2 4 For each of the following functions, whose x-value is a multiple of g ( x ) = tan ( x ) , where . For 1 34. f ( x ) = 5 tan x + 2 4 4 35. f ( x ) = 3cot x + + 2 2 36. f ( x ) = 4 cot x + 3 3 consistency in solutions, perform the Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions Answer the following. 37. The graph of f ( x ) = sin ( x ) can be useful in sketching transformations of the graph of g ( x ) = csc ( x ) . Answer the following, using the interval 2 x 2 for each graph. (a) Sketch the graphs of f ( x ) = sin ( x ) and g ( x ) = csc ( x ) on the same set of axes. (b) Where are the x-intercepts of f ( x ) = sin ( x ) ? (c) Where are the asymptotes of g ( x ) = csc ( x ) ? Explain the relationship between the answers in parts (b) and (c). (d) On a different set of axes, sketch the graph of h ( x ) = 2sin ( 3 x ) . (e) Use the graph in part (d) to sketch the graph of p ( x ) = 2 csc ( 3 x ) on the same set of axes. (f) On a different set of axes, sketch the graph of q ( x ) = 4sin x 3 . 2 (g) Use the graph in part (f) to sketch the graph of r ( x ) = 4 csc x 3 on the same set 2 of axes. 38. The graph of f ( x ) = cos ( x ) can be useful in sketching transformations of the graph of g ( x ) = sec ( x ) . Answer the following, using the interval 2 x 2 for each graph. (a) Sketch the graphs of f ( x ) = cos ( x ) and g ( x ) = sec ( x ) on the same set of axes. (b) Where are the x-intercepts of f ( x ) = cos ( x ) ? (c) Where are the asymptotes of g ( x ) = sec ( x ) ? Explain the relationship between the answers in parts (b) and (c). (d) On a different set of axes, sketch the graph of h ( x ) = 3cos ( 2 x ) + 1 . Math 1330, Precalculus The University of Houston (e) Use the graph in part (d) to sketch the graph of p ( x ) = 3sec ( 2 x ) + 1 on the same set of axes. (f) On a different set of axes, sketch the graph of q ( x ) = 5cos ( 2 x + ) . (g) Use the graph in part (f) to sketch the graph of r ( x ) = 5sec ( 2 x + ) on the same set of axes. For each of the following functions, (a) State the period. (b) Use transformations to sketch the graph of the function over one period. The x-axis should be labeled to reflect the location of any x-values where a relative maximum or minimum occurs. The y-axis should be labeled to reflect any maximum and minimum values of the function. Label any asymptotes clearly. For consistency in solutions, perform the appropriate transformations on g ( x ) = sec ( x ) where 0 x 2 , or h ( x ) = csc ( x ) , where 0 < x < 2 . (Note: The resulting graph may not fall within these intervals.) 39. f ( x ) = 8sec ( x ) 40. f ( x ) = 5csc ( x ) 41. f ( x ) = 4 csc ( x ) + 5 42. f ( x ) = 2sec ( x ) 3 x 43. f ( x ) = sec 2 2 x 44. f ( x ) = sec 3 5 45. f ( x ) = 7 sec ( 3x ) 1 46. f ( x ) = 4 csc ( 2 x ) + 3 47. f ( x ) = 7 csc ( 2 x ) + 1 2 48. f ( x ) = 2 csc ( x ) 3 Chapter 5: Trigonometric Functions Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions x 49. f ( x ) = 4 csc +5 3 60. y 2 x 50. f ( x ) = 2sec 6 4 3/4 /2 /4 x /4 2 /2 3/4 4 6 51. f ( x ) = 4sec ( x ) 2 8 10 52. f ( x ) = 8csc x + + 3 2 61. 53. f ( x ) = 2 csc x + 2 2 5/8 /2 3/8 /4 /8 2 54. f ( x ) = 4 csc ( 2 x ) 4 55. f ( x ) = 3sec ( 3x ) 8 y x /8 /4 3/8 /2 6 10 12 56. f ( x ) = 9sec ( 3 2 x ) x 57. f ( x ) = 2 csc + 4 3 2 62. 8 y 6 4 x 2 + 58. f ( x ) = 9sec +2 3 6 2 /3 /4 /6 /12 2 x /12 /6 4 For each of the following graphs, (a) Give an equation of the form f ( x ) = A tan ( Bx C ) + D which could be For each of the following graphs, used to represent the graph. (Note: C or D may be zero. Answers vary.) (a) Give an equation of the form f ( x ) = A sec ( Bx C ) + D which could be (b) Give an equation of the form f ( x ) = A cot ( Bx C ) + D which could be used to represent the graph. (Note: C or D may be zero. Answers vary.) used to represent the graph. (Note: C or D may be zero. Answers vary.) 59. (b) Give an equation of the form f ( x ) = A csc ( Bx C ) + D which could be y f 8 used to represent the graph. (Note: C or D may be zero. Answers vary.) 6 4 63. 2 3/4 /2 /4 2 4 6 4 x /4 /2 3/4 y 2 3/2 /2 2 x /2 3/2 2 4 6 8 Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions 64. 10 y 8 6 4 2 /2 x /2 2 4 65. 10 y 8 6 4 2 x 5 4 3 2 1 1 2 3 2 4 66. y 2 2.0 1.5 1.0 0.5 2 x 0.5 1.0 1.5 2.0 2.5 3.0 4 6 8 10 12 Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions Exercises 1-6 help to establish the graphs of the inverse trigonometric functions, and part (h) of each exercise shows a shortcut for remembering the range. Answer the following. 1. (d) Plot the points from part (c) on the axes below. Then use those points along with the graph from part (b) to sketch the graph of x = cos ( y ) . (a) Complete the following chart. Round to the nearest hundredth. x y = cos ( x ) 2 3/2 y = cos ( x ) x 0 /2 2 x 2 1.2 1.0 0.8 0.6 0.4 0.2 2 2 /2 0.2 0.4 y x /2 3/2 1.2 (f) Using the graph of x = cos ( y ) from part (d), sketch the portion of the graph with each of the following restricted ranges. Then state if each graph represents a function. 3 2, 2 i. [0, ] ii. iii. [ , 0] iv. , 2 2 (g) One of the functions from part (f) is the graph of f ( x ) = cos 1 ( x ) . Which one? 1.2 (c) Use the chart from part (a) to complete the following chart for x = cos ( y ) . y 1.0 (e) Is the inverse relation in part (d) a function? Why or why not? 2 0.6 0.8 1.0 x = cos ( y ) 0.8 2 0.4 0.2 0.6 3/2 1.0 0.8 0.6 3/2 0.4 3 2 (b) Plot the points from part (a) on the axes below. Then use those points along with previous knowledge of trigonometric graphs to sketch the graph of y = cos ( x ) . 1.2 0.2 /2 3 2 2 y x = cos ( y ) y 0 2 2 3 2 2 2 Continued in the next column... 3 2 Explain why that particular range may be the most reasonable choice. (h) Below is a way to remember the range of f ( x ) = cos 1 ( x ) without drawing a graph: i. Draw a unit circle on the coordinate plane. In each quadrant, write \" + \" if the cosine of angles in that quadrant are positive, and write \" \" if the cosine of angles in that quadrant are negative. ii. Using the diagram from part i, find two consecutive quadrants where one is labeled \" + \" and the other is labeled \" \". Write down any other pairs of consecutive quadrants where this occurs. Continued on the next page... Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions iii. Of the answers from part ii, choose the pair of quadrants that includes Quadrant I; this illustrates the range of f ( x ) = cos 1 ( x ) . Write the range in (d) Plot the points from part (c) on the axes below. Then use those points along with the graph from part (b) to sketch the graph of x = sin ( y ) . interval notation. 2. 2 3/2 (a) Complete the following chart. Round to the nearest hundredth. x y = sin ( x ) y = sin ( x ) x y /2 x 0 1.2 1.0 0.8 0.6 0.4 0.2 2 3 2 3 2 2 2 0.4 0.6 0.8 1.0 1.2 /2 2 0.2 3/2 2 (e) Is the inverse relation in part (d) a function? Why or why not? (b) Plot the points from part (a) on the axes below. Then use those points along with previous knowledge of trigonometric graphs to sketch the graph of y = sin ( x ) . 1.2 y 1.0 0.8 0.6 0.4 0.2 2 3/2 /2 0.2 0.4 x /2 3/2 2 0.6 0.8 1.0 (c) Use the chart from part (a) to complete the following chart for x = sin ( y ) . y sketch the portion of the graph with each of the following restricted ranges. Then state if each graph represents a function. 3 2, 2 i. [0, ] ii. iii. [ , 0] iv. , 2 2 (g) One of the functions from part (f) is the graph of f ( x ) = sin 1 ( x ) . Which one? 1.2 x = sin ( y ) (f) Using the graph of x = sin ( y ) from part (d), x = sin ( y ) y 2 2 3 2 2 2 Continued in the next column... (h) Below is a way to remember the range of f ( x ) = sin 1 ( x ) without drawing a graph: i. 0 Explain why that particular range may be the most reasonable choice. 3 2 Draw a unit circle on the coordinate plane. In each quadrant, write \" + \" if the sine of angles in that quadrant are positive, and write \" \" if the sine of angles in that quadrant are negative. ii. Using the diagram from part i, find two consecutive quadrants where one is labeled \" + \" and the other is labeled \" \". Write down any other pairs of consecutive quadrants where this occurs. Continued on the next page... Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions iii. Of the answers from part ii, choose the pair of quadrants that includes Quadrant I; this illustrates the range of f ( x ) = sin 1 ( x ) . Write the range in (d) Plot the points from part (c) on the axes below. Then use those points along with the graph from part (b) to sketch the graph of x = tan ( y ) . interval notation. 3. 3/4 (a) Complete the following chart. Round to the nearest hundredth. If a value is undefined, state \"Undefined.\" x y = tan ( x ) y = tan ( x ) x /2 /4 x 1.2 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 /4 4 2 /2 4 3/4 2 3 4 3 4 (e) Is the inverse relation in part (d) a function? Why or why not? (b) Plot the points from part (a) on the axes below. Then use those points along with previous knowledge of trigonometric graphs to sketch the graph of y = tan ( x ) . 1.2 0.4 0.2 3/4 /2 /4 0.2 0.4 x /4 /2 3/4 0.6 0.8 1.0 1.2 (c) Use the chart from part (a) to complete the following chart for x = tan ( y ) . x = tan ( y ) y (f) Using the graph of x = tan ( y ) from part (d), sketch the portion of the graph with each of the following restricted ranges. Then state if each graph represents a function. 0, , 2 2 ii. , , 0 2 2 iii. 0, , 2 2 iv. , 2 2 i. y 1.0 0.8 0.6 y x = tan ( y ) y 0 4 2 4 2 3 4 3 4 Continued in the next column... (g) One of the functions from part (f) is the graph of f ( x ) = tan 1 x . Which one? Explain why that particular range may be the most reasonable choice. (h) Below is a way to remember the range of f ( x ) = tan 1 ( x ) without drawing a graph: i. Draw a unit circle on the coordinate plane. In each quadrant, write \" + \" if the tangent of angles in that quadrant are positive, and write \" \" if the tangent of angles in that quadrant are negative. Also make a note next to any quadrantal angles for which the tangent function is undefined. Continued on the next page... Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions ii. Using the diagram from part i, find two consecutive quadrants where one is labeled \" + \" and the other is labeled \" \". Write down any other pairs of consecutive quadrants where this occurs. iii. Of the answers from part ii, write down any pairs of quadrants that include Quadrant I. (There are two pair.) Suppose for a moment that either pair could illustrate the range of f ( x ) = tan 1 ( x ) , and write down the ranges for each in interval notation, remembering to consider where the function is undefined. Then choose the range that is represented by a single interval. This is the range of f ( x ) = tan 1 ( x ) . (c) Use the chart from part (a) to complete the following chart for x = cot ( y ) . x = cot ( y ) x = cot ( y ) y 0 4 2 y = cot ( x ) y /2 /4 x 0 1.2 1.0 0.8 0.6 0.4 0.2 4 2 0.2 2 3/4 y 1.0 0.8 0.6 0.4 0.2 /2 /4 0.2 0.4 x /4 0.6 0.8 1.0 1.2 Continued in the next column... Math 1330, Precalculus The University of Houston 0.8 1.0 1.2 (e) Is the inverse relation in part (d) a function? Why or why not? (b) Plot the points from part (a) on the axes below. Then use those points along with previous knowledge of trigonometric graphs to sketch the graph of y = cot ( x ) . 3/4 0.6 /2 3 4 1.2 0.4 /4 4 3 4 2 3 4 3/4 y = cot ( x ) x 4 (d) Plot the points from part (c) on the axes below. Then use those points along with the graph from part (b) to sketch the graph of x = cot ( y ) . (a) Complete the following chart. Round to the nearest hundredth. If a value is undefined, state \"Undefined.\" x 3 4 4. y /2 3/4 (f) Using the graph of x = cot ( y ) from part (d), sketch the portion of the graph with each of the following restricted ranges. Then state if each graph represents a function. , 0 0, 2 2 i. ( 0, ) ii. iii. ( , 0 ) iv. , 0 0, 2 2 (g) One of the functions from part (f) is the graph of f ( x ) = cot 1 x . Which one? Explain why that particular range may be the most reasonable choice. Continued on the next page... Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions (h) Below is a way to remember the range of f ( x ) = cot 1 ( x ) without drawing a graph: i. Draw a simple coordinate plane by sketching the x and y-axes. In each quadrant, write \" + \" if the cotangent of numbers in that quadrant are positive, and write \" \" if the cotangent of numbers in that quadrant are negative. Also make a note next to any quadrantal angles for which the cotangent function is undefined. ii. Using the diagram from part i, find two consecutive quadrants where one is labeled \" + \" and the other is labeled \" \". Write down any other pairs of consecutive quadrants where this occurs. iii. Of the answers from part ii, write down any pairs of quadrants that include Quadrant I. (There are two pair.) Suppose for a moment that either pair could illustrate the range of f ( x ) = cot 1 ( x ) , and write down the ranges for each in interval notation, remembering to consider where the function is undefined. Then choose the range that is represented by a single interval. This is the range of f ( x ) = cot 1 ( x ) . 5. (a) Complete the following chart. Round to the nearest hundredth. If a value is undefined, state \"Undefined.\" x y = csc ( x ) x 2 4 y 3 2 1 2 3/2 /2 x /2 1 3/2 2 2 3 4 (c) Use the chart from part (a) to complete the following chart for x = csc ( y ) . x = csc ( y ) x = csc ( y ) y y 0 2 2 3 2 3 2 2 2 (d) Plot the points from part (c) on the axes below. Then use those points along with the graph from part (b) to sketch the graph of x = csc ( y ) . 2 y = csc ( x ) y 3/2 0 (b) Plot the points from part (a) on the axes below. Then use those points along with previous knowledge of trigonometric graphs to sketch the graph of y = csc ( x ) . 2 3 2 3 2 2 2 /2 x 1.2 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 1.2 /2 3/2 2 Continued in the next column... (e) Is the inverse relation in part (d) a function? Why or why not? Continued on the next page... Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions (f) Using the graph of x = csc ( y ) from part (d), sketch the portion of the graph with each of the following restricted ranges. Then state if each graph represents a function. i. iii. (b) Plot the points from part (a) on the axes below. Then use those points along with previous knowledge of trigonometric graphs to sketch the graph of y = sec ( x ) . ( 0, ) ii. 3 2 , , 2 4 ( , 0 ) iv. , 0 0, 2 2 1 2 2 3/2 /2 (h) Below is a way to remember the range of f ( x ) = csc 1 ( x ) without drawing a graph: i. Draw a unit circle on the coordinate plane. In each quadrant, write \" + \" if the cosecant of angles in that quadrant are positive, and write \" \" if the cosecant of angles in that quadrant are negative. Also make a note next to any quadrantal angles for which the cosecant function is undefined. ii. Using the diagram from part i, find two consecutive quadrants where one is labeled \" + \" and the other is labeled \" \". Write down any other pairs of consecutive quadrants where this occurs. iii. Of the answers from part ii, choose the pair of quadrants that includes Quadrant I; this illustrates the range of f ( x ) = csc 1 ( x ) . Consider where the y = sec ( x ) x 0 2 2 3 4 (c) Use the chart from part (a) to complete the following chart for x = sec ( y ) . x = sec ( y ) x = sec ( y ) y y 0 2 2 3 2 3 2 2 2 (d) Plot the points from part (c) on the axes below. Then use those points along with the graph from part (b) to sketch the graph of x = sec ( y ) . 2 y 3/2 x y = sec ( x ) 1.2 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 1.2 /2 2 3/2 2 3 2 2 2 3 2 Continued in the next column... Math 1330, Precalculus The University of Houston 3/2 /2 (a) Complete the following chart. x 2 function is undefined, and then write the range in interval notation. 6. x /2 1 (g) One of the functions from part (f) is the graph of f ( x ) = csc1 ( x ) . Which one? Explain why that particular range may be the most reasonable choice. y 3 (e) Is the inverse relation in part (d) a function? Why or why not? Continued on the next page... Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions (f) Using the graph of x = sec ( y ) from part (d), sketch the portion of the graph with each of the following restricted ranges. Then state if each graph represents a function. i. 0, 2 2 , ii. 3 , 2 2 iii. , , 0 2 2 iv. , 2 2 (g) One of the functions from part (f) is the graph of f ( x ) = sec 1 ( x ) . Which one? Explain why that particular range may be the most reasonable choice. (h) Below is a way to remember the range of f ( x ) = sec 1 ( x ) without drawing a graph: i. Draw a unit circle on the coordinate plane. In each quadrant, write \" + \" if the secant of angles in that quadrant are positive, and write \" \" if the secant of angles in that quadrant are negative. Also make a note next to any quadrantal angles for which the secant function is undefined. ii. Using the diagram from part i, find two consecutive quadrants where one is labeled \" + \" and the other is labeled \" \". Write down any other pairs of consecutive quadrants where this occurs. iii. Of the answers from part ii, choose the pair of quadrants that includes Quadrant I; this illustrates the range of f ( x ) = sec 1 ( x ) . Consider where the function is undefined, and then write the range in interval notation. Inverse functions may be easier to remember if they are translated into words. For example: sin 1 ( x ) = the number (in the interval 2 , 2 ) whose sine is x. Translate each of the following expressions into words (and include the range of the inverse function in parentheses as above). Then find the exact value of each expression. Do not use a calculator. 7. 1 (a) cos 1 2 (b) tan 1 (1) 8. 2 (a) sin 1 2 (b) cot 1 ( 1) 9. (a) sin 1 ( 1) (b) sec 1 3 10. (a) cos 1 2 (b) csc1 ( 2 ) 11. sin sin 1 ( 0.2 ) 12. cos arccos ( 0.7 ) 13. tan arctan ( 4 ) 14. csc csc1 ( 3.5 ) Find the exact value of each of the following expressions. Do not use a calculator. If undefined, state, \"Undefined.\" 1 15. (a) sin 1 2 (b) cos 1 (1) 3 16. (a) cos 1 2 (b) arcsin (1) 17. (a) arccos ( 3) (b) sin 1 ( 0 ) ( ) 18. (a) sin 1 2 Math 1330, Precalculus The University of Houston ( 2) (b) cos 1 ( 1) ( 3) 19. (a) tan 1 ( 1) (b) tan 1 20. (a) arctan ( 0 ) 3 (b) tan 1 3 Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions 3 21. (a) cot 1 3 (b) cot 1 ( 0 ) 22. (a) cot 1 (1) (b) cot 1 3 23. (a) sec 1 ( 2 ) (b) arccsc ( 1) ( Use a calculator to find the value of each of the following expressions. Round each answer to the nearest thousandth. If undefined, state, \"Undefined.\" ) 2 3 24. (a) csc 3 (b) sec 2 25. (a) csc1 2 2 3 (b) sec 1 3 1 26. (a) sec 1 1 ( 0) 27. (a) sec ( 2 ) ( x) = 1 1 (b) sec 2 obtained by rounding later in the calculation: 1 tan 1 2.697 . (Either answer is acceptable.) 2.1 4 33. (a) sin 1 7 7 (b) arccos 9 34. (a) cos 1 ( 3) 1 (b) sin 1 5 35. (a) tan 1 ( 4 ) (b) arctan ( 0.12 ) 36. (a) arctan ( 0.28 ) (b) tan 1 ( 30 ) 1 3 37. (a) cot 1 8 (b) cot 1 ( 0.8 ) 1 38. (a) cot 1 ( 6 ) 11 (b) cot 1 5 1 39. (a) cot 1 5 (b) arccot ( 10 ) 40. (a) cot 1 ( 7 ) (b) cot 1 ( .62 ) 41. (a) csc1 ( 2.4 ) 10 (b) sec 1 3 1 28. (a) csc1 ( x ) = sin 1 x 1 1 ( x) 1 29. (a) cot ( x ) = tan x (b) cot 1 ( x ) = cot 1 ( 2.1) 0.444 2.698 . A more exact answer can be (b) cos 1 ( 0.24 ) 1 1 number, the solution lies in the interval , . Therefore, 2 32. (a) sin 1 ( 0.9 ) cos 1 ( x ) sin 2.1 .\" Since the range of the inverse cotangent function is 2.1 , a negative (b) sin 1 ( 0.8 ) 1 (b) sec ( x ) = cos x (b) csc1 ( x ) = cot 1 ( 2.1) can be translated as \"the cotangent of a number is 31. (a) cos 1 ( 0.37 ) 1 1 1 reference angle) by finding tan 1 0.444 . The expression 2.1 ( 0, ) , and the cotangent of the number is Answer True or False. Assume that all x-values are in the domain of the given inverse functions. 1 Note: To find the inverse cotangent of a negative number, such as cot 1 ( 2.1) , first find a reference number (to be likened to a tan 1 ( x ) 1 30. Explain why cot 1 ( x ) tan 1 . Are there x 1 any values for which cot 1 ( x ) = tan 1 ? If x so, which values? Math 1330, Precalculus The University of Houston Chapter 5: Trigonometric Functions Exercise Set 5.4: Inverse Trigonometric Functions 3 42. (a) sec1 4 (b) csc1 ( 7.2 ) 43. (a) sec1 ( 0.71) 31 (b) csc1 5 44. (a) csc1 ( 9.5 ) (b) arcsec ( 8.8 ) 3 56. (a) tan cos 1 2 ( 3 57. (a) csc tan 1 3 Find the exact value of each of the following expressions. Do not use a calculator. If undefined, state, \"Undefined.\" ( 3 ) 45. (a) sin sin 1 ( 0.84 ) (b) cos cos 1 46. (a) sin sin 1 ( 5 ) 2 (b) cos cos 1 3 47. (a) csc csc1 ( 3.5 ) (b) cot cot 1 ( 7 ) 48. (a) sec sec 1 ( 0.3) (b) tan tan 1 ( 0.5 ) 3 49. (a) sin cos 1 5 8 (b) tan sin 1 17 12 50. (a) cos tan 1 5 7 (b) cot cos 1 25 2 51. (a) sec tan 1 7 (b) cot csc1 ( 5 ) 52. (a) csc cot 1 ( 4 ) 7 (b) tan sec1 4 12 53. (a) sin cos 1 13 1 (b) tan cos 1 6 1 54. (a) cos tan 1 7 3 (b) cot sin 1 7 55. (a) sin tan 1 ( 1) 2 (b) cos sin 1 2 Math 1330, Precalculus The University of Houston ) (b) cos tan 1 3 (b) tan sec 1 ( 2 ) 1 58. (a) cot cos 1 2 ( ) (b) sin csc 1 2 4 59. (a) cos 1 cos 3 (b) sin 1 sin 4 60. (a) cos 1 cos 6 2 (b) sin 1 sin 3 7 61. (a) sec1 sec 6 3 (b) tan 1 tan 4 7 62. (a) cot 1 cot 4 4 (b) csc 1 csc 3 Sketch the graph of each function. 63. y = cos 1 ( x 2 ) 64. y = arctan ( x + 1) 65. y = cot 1 ( x ) 2 66. y = csc 1 ( x ) 67. y = arcsin ( x 1) + 68. y = cos 1 ( x + 2 ) + 2 Chapter 5: Trigonometric Functions Math 1330 Precalculus Electronic Homework (EHW 7) Sections 5.3, 5.4. Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in the Online Textbook under the Exercises link in the appropriate section. Record your answers by logging into your CASA account, go to the EMCF link in the top menu bar, selecting the appropriate assignment in the EMCF menu and recording your choices on the EMCF form. You MUST record your answers by the deadline given. No late homework will be accepted. NOTE: This homework has 20 questions; your grade will be out of 20. Problem 5.3.12 refers to problem 12 in Chapter 5, Section 3 in the online text. 1. Problem 5.3.26: Which of these is an asymptote of the graph? (Note restrictions.) x A. E. x B. 4 x C. 3 x D. 6 x 2 F. None of these 2 2. Problem 5.3.34 a A. E. B. 4 2 C. 4 D. 2 F. None of these Page 1 of 9 3. Problem 5.3.32: Which is the graph of the function? A. B. C. D. E. None of these Page 2 of 9 4. Problem 5.3.34: Which is the graph of the function? Pay attention to the scale of the graph. A. B. C. D. E. None of these Page 3 of 9 5. Problem 5.3.60: Write an equation of the function that is graphed. A. f ( x) 2 tan x B. f ( x) C. f ( x) D. f ( x) 2 cot x E. None of the above 4 2 2 cot x 4 2 tan x 4 4 2 6. Problem 5.3.50: Which of these gives the period of the function? A. E. B. 4 8 C. 2 8 D. 1 2 F. None of these 7. Problem 5.3.64: Write an equation of the function that is graphed. A. f ( x) B. f ( x ) sec x C. f ( x) D. f ( x) 2 sec x E. f ( x) csc x F. 2sec x 3 2 4 2sec x 3 3 4 None of the above Page 4 of 9 Section 5.4: 8. The range of f x 0, A. E. sin B. 0, 1 2 x is: , , C. 2 1 ,1 D. F. None of these 9. The range of f x 0, A. 1 x is: B. 1 ,1 D. tan 2 , 0, E. , C. 2 F. None of these 10. Problem 5.4.20 b A. B. 6 D. F. 5 6 C. E. 6 11 6 3 None of these 11. Problem 5.4.26 a A. D. Undefined 3 4 B. E. 4 5 6 C. 3 4 F. None of these Page 5 of 9 3 4 A. 1 2 arcsin 12. Evaluate: B. arccos C. 4 5 4 1 2 arctan D. 4 1 E. 5 12 F. None of these 13. Problem 5.4.46 A. -5; 2/3 B. 5; -2/3 C. doesn't exist; 2/3 D. -5; doesn't exist E. doesn't exist; doesn't exist F. None of these 14. Problem 5.4.50 A. 5 ; 13 24 7 B. 12 ; 5 C. 5 ; 13 7 24 D. 5 119 ; 119 7 25 E. None of the above 674 7 Page 6 of 9 15. Problem 5.4.52 A. 17 ; 33 4 B. 17 ; 4 33 33 C. 15 ; 7 4 D. 17 ; 4 E. 33 7 None of the above 16. Problem 5.4.54 A. B. C. 3 10 20 5 2 ; 7 2 10 3 2; 10 3 10 20 7 2 ; 10 2 10 3 D. 7 2; 10 E. None of the above 17. Evaluate: A. 11 3 cos B. 1 4 3 cos 2 sin 1 sin C. 7 6 2 Page 7 of 9 D. 7 5 6 E. 6 F. None of these 18. Problem 5.4.64: Which of the following points lies on the graph of the function that is given? A. E. 4 ,0 2, 3 4 B. 0, 4 C. 2, 4 D. 4 , 2 F. None of these 19. The graph of a function is given below. Which of the following can be this function? A. f ( x) arcsin( x 1) 2 B. f ( x) arccos( x 2) 1 C. f ( x) 2 arcsin( x 2) D. f ( x) arcsin( x 2) E. f ( x) arcsin( x 2) 2 F. None of these Page 8 of 9 20. Let y A. arcsin 1 x2 E. 2 4 x2 x . Find an expression for cos( y ) in terms of \"x\". 2 B. 2 4 x 2 C. 4 x2 2 D. 4 x2 F. None of these Page 9 of 9