Graph each function in the interval from 0 to 2π.

43. y = csc θ − π 2 y equals co-secant theta minus , pi over 2
44. y = sec 1 4 θ y equals secant , 1 fourth , theta
45. y = − sec π θ y equals negative secant pi theta
46. y = cot θ 3 y equals co-tangent , theta over 3
47. 1. What are the domain, range, and period of y = csc x ? y equals co-secant x question mark
    2. What is the relative minimum in the interval 0 ≤ x ≤ π ? 0 less than or equal to x less than or equal to pi question mark
    3. What is the relative maximum in the interval π ≤ x ≤ 2 π ? pi less than or equal to x less than or equal to 2 pi question mark
48. Reasoning Use the relationship csc x = 1 sin x co-secant x equals . fraction 1 , over sine x end fraction  to explain why each statement is true.
    1. When the graph of y = sin x y equals sine x  is above the x-axis, so is the graph of y = csc x . y equals co-secant x .
    2. When the graph of y = sin x y equals sine x  is near a y-value of − 1 , negative 1 comma  so is the graph of y = csc x . y equals co-secant x .

Writing Explain why each expression is undefined.

49. csc 180 ° co-secant , 180 degrees
50. sec 90 ° secant , 90 degrees
51. cot 0 ° co-tangent 0 degrees

52. Indirect Measurement The fire ladder forms an angle of measure θ with the horizontal. The hinge of the ladder is 35 ft from the building. The function y = 35 sec θ y equals 35 secant theta  models the length y in feet that the fire ladder must be to reach the building.

    

    1. Graph the function.
    2. In the photo, θ = 13 ° . theta equals 13 degrees .  What is the ladder's length?
    3. How far is the ladder extended when it forms an angle of 30°?
    4. Suppose the ladder is extended to its full length of 80 ft. What angle does it form with the horizontal? How far up a building can the ladder reach when fully extended? (Hint: Use the information in the photo.)
53. 1. Graph y = tan x y equals tangent x  and y = cot x y equals co-tangent x  on the same axes.
    2. State the domain, range, and asymptotes of each function.
    3. Compare and Contrast Compare the two graphs. How are they alike? How are they different?
    4. Geometry The graph of the tangent function is a reflection image of the graph of the cotangent function. Name at least two reflection lines for such a transformation.

Graphing Calculator Graph each function in the interval from 0 to 2π. Describe any phase shift and vertical shift in the graph.

54. y = sec 2 θ + 3 y equals secant 2 theta plus 3
55. y = sec 2 ( θ + π 2 ) y equals secant 2 . open . theta plus , pi over 2 . close
56. y = − 2 sec ( x − 4 ) y equals negative 2 secant open x minus 4 close
57. f ( x ) = 3 csc ( x + 2 ) − 1 f open x close equals 3 csc open x plus 2 close minus 1
58. y = cot 2 ( x + π ) + 3 y equals co-tangent 2 open x plus pi close plus 3
59. g ( x ) = 2 sec ( 3 ( x − π 6 ) ) − 2 g , open x close , equals 2 secant . open . 3 . open . x minus , pi over 6 . close . close . minus 2
60. 1. Graph y = − cos x y equals negative cosine x  and y = − sec x y equals negative secant x  on the same axes.
    2. State the domain, range, and period of each function.
    3. For which values of x does − cos x = − sec x ? negative cosine x equals negative secant x question mark  Justify your answer.
    4. Compare and Contrast Compare the two graphs. How are they alike? How are they different?
    5. Reasoning Is the value of − sec x negative secant x  positive when − cos x negative cosine x  is positive and negative when − cos x negative cosine x  is negative? Justify your answer.

Table of Contents