61. 1. Reasoning Which expression gives the correct value of csc 60°?
       1. sin ( ( 60 − 1 ) ° ) sine open open , 60 super negative 1 end super , close degrees close
       2. ( sin 60 ° ) − 1 open sine , 60 degrees close super negative 1 end super
       3. ( cos 60 ° ) − 1 open cosine , 60 degrees close super negative 1 end super
    2. Which expression in part (a) represents sin ( 1 60 ) ∘ ? sine . open , 1 sixteth , close to the composition . question mark

C Challenge

62. Reasoning Each branch of y = sec x y equals secant x  and y = csc x y equals csc x  is a curve. Explain why these curves cannot be parabolas. (Hint: Do parabolas have asymptotes?)
63. Reasoning Consider the relationship between the graphs of y = cos x y equals cosine x  and y = cos 3 x . y equals cosine 3 x .  Use the relationship to explain the distance between successive branches of the graphs of y = sec x y equals secant x  and y = sec 3 x . y equals secant 3 x .
64. 1. Graph y = cot x , y = cot 2 x , y = cot ( − 2 x ) , y equals co-tangent x comma y equals co-tangent 2 x comma y equals co-tangent open negative 2 x close comma  and y = cot 1 2 x y equals co-tangent , 1 half , x  on the same axes.
    2. Make a Conjecture Describe how the graph of y = cot b x y equals co-tangent b x  changes as the value of b changes.

Standardized Test Prep

GRIDDED RESPONSE

SAT/ACT

For Exercises 65–68, suppose cos θ = 3 5 cosine theta equals , 3 fifths  and sin θ > 0 . sine theta greater than 0 .  Enter each answer as a fraction.

65. What is tan θ?
66. What is sec θ?
67. What is cot θ?
68. What is csc θ?

For Exercises 69–70, suppose tan θ = 4 3 theta equals , 4 thirds  and − π 2 ≤ θ < π 2 . negative , pi over 2 , less than or equal to theta less than , pi over 2 . .  Enter each answer as a decimal. Round your answer to the nearest tenth, if necessary.

69. What is cot θ + cos θ ? co-tangent theta plus cosine theta question mark
70. What is ( sin θ ) ( cot θ ) ? open sine theta close open co-tangent theta close question mark

Mixed Review

Find the amplitude and period of each function. Describe any phase shift and vertical shift in the graph. See Lesson 13-7.

71. y = 2 sin x − 5 y equals 2 sine x minus 5
72. y = − cos ( x + 4 ) − 7 y equals negative cosine open x plus 4 close minus 7
73. y = − 3 sin ( x + π 6 ) + 4 y equals negative 3 sine . open . x plus , pi over 6 . close . plus 4
74. y = 5 cos π ( x − 1 . 5 ) − 8 y equals 5 cosine pi open x minus 1 . 5 close minus 8

Sketch a normal curve for each distribution. Label the x-axis values at one, two, and three standard deviations from the mean. See Lesson 11-9.

75. mean = 25, standard deviation = 5
76. mean = 25, standard deviation = 10

Get Ready! To prepare for Lesson 14-1, do Exercises 77–79.

Determine whether each equation is true for all real numbers x. Explain your reasoning. See Lesson 1-4.

77. 2 x + 3 x = 5 x 2 x plus 3 x equals 5 x
78. − ( 4 x − 10 ) = 10 − 4 x negative open 4 x minus 10 close equals 10 minus 4 x
79. 3 x + 15 = 5 ( x − 3 ) − 2 x 3 x plus 15 equals 5 open x minus 3 close minus 2 x

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