P.6 Assess Your Understanding

Concepts and Vocabulary

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The sine, cosine, cosecant, and secant functions have period _____; the tangent and cotangent functions have period _____.

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The domain of the tangent function \(f( x) = \tan x\) is _____.

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The range of the sine function \(f( x) = \sin x\) is _____.

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Explain why \(\tan\! \left(\! \dfrac{\pi }{4}+2\pi \right) =\tan \dfrac{\pi }{4}\).

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True or False  The range of the secant function is the set of all positive real numbers.

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The function \(f( x) = 3\cos (6x)\) has amplitude _____ and period _____.

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True or False  The graphs of \(y=\sin x\) and \(y=\cos x\) are identical except for a horizontal shift.

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True or False  The amplitude of the function \(f( x) =2\sin ( \pi x)\) is \(2\) and its period is \(\dfrac{\pi }{2}\).

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True or False  The graph of the sine function has infinitely many \(x\)-intercepts.

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The graph of \(y=\tan x\) is symmetric with respect to the _____.

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The graph of \(y=\sec x\) is symmetric with respect to the _____.

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Explain, in your own words, what it means for a function to be periodic.

Practice Problems

In Problems 13–16, use the even-odd properties to find the exact value of each expression.

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\(\tan\! \left(\! -\dfrac{\pi }{4} \right)\)

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\(\sin\! \left(\!\! -\dfrac{3\pi }{2}\! \right)\)

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\(\csc\! \left(\! -\dfrac{\pi }{3} \right)\)

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\(\cos\! \left(\! -\dfrac{\pi }{6} \right)\)

57

In Problems 17–20, if necessary, refer to a graph to answer each question.

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What is the \(y\)-intercept of \({f}( x) =\tan x\)?

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Find the \(x\)-intercepts of \(f( x) =\sin x\) on the interval \([-2\pi ,2\pi]\).

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What is the smallest value of \(f(x) =\cos x\)?

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For what numbers \(x\), \(-2\pi \leq x\leq 2\pi\), does \(\sin x=1\)? Where in the interval \([ -2\pi ,2\pi]\) does \(\sin x=-1\)?

In Problems 21–26, the graphs of six trigonometric functions are given. Match each graph to one of the following functions:

  1. \(y=2\sin\! \left( \dfrac{\pi }{2}x\right)\)
  2. \(y=2\cos\! \left( \dfrac{\pi }{2}x\right)\)
  3. \(y=3\cos ( 2x)\)
  4. \(y=-3\sin ( 2x)\)
  5. \(y=-2\cos\! \left( \dfrac{\pi }{2}x\right)\)
  6. \(y=-2\sin\! \left( \dfrac{1}{2}x\right)\)

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In Problems 27–32, graph each function using transformations. Be sure to label key points and show at least two periods.

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\(f( x) =4\sin (\pi x)\)

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\(f( x) =-3\cos x\)

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\(f( x) =3\cos ( 2x) -4\)

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\(f( x) =4\sin ( 2x) +2\)

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\(f( x) =\tan\! \left( \dfrac{\pi }{2}x\! \right) \)

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\(f( x) =4\sec\! \left( \dfrac{1}{2}x\! \right)\)

In Problems 33–36, determine the amplitude and period of each function.

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\(g( x) =\dfrac{1}{2}\cos ( \pi x)\)

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\(f( x) =\sin ( 2x)\)

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\(g( x) =3\sin x\)

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\(f( x) =-2\cos\! \left( \dfrac{3}{2}x\!\right)\)

In Problems 37 and 38, write the sine function that has the given properties.

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Amplitude: 2, Period: \(\pi\)

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Amplitude: \(\dfrac{1}{3}\), Period: 2

In Problems 39 and 40, write the cosine function that has the given properties.

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Amplitude: \(\dfrac{1}{2}\), Period: \(\pi\)

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Amplitude: \(3\), Period: \(4\pi\)

In Problems 41–48, for each graph, find an equation involving the indicated trigonometric function.

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