P.6 Assess Your Understanding

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Concepts and Vocabulary

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

\(2\pi\), \(\pi\)

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

All real numbers except odd multiples of \(\frac{\pi}{2}\)

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

\(\{y | -1 \le y \le 1\}\)

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

The period of \(\tan x\) is \(\pi\).

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

False

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

\(3\), \(\frac{\pi}{3}\)

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

True

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

False

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

True

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

Origin

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

\(y\)-axis

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

Answers will vary.

Practice Problems

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

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

\(-1\)

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

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

\(-\frac{2\sqrt{3}}{3}\)

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

57

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

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

\(0\)

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

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

\(-1\)

  1. 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. (a) \(y=2\sin\! \left( \dfrac{\pi }{2}x\right)\)
  2. (b) \(y=2\cos\! \left( \dfrac{\pi }{2}x\right)\)
  3. (c) \(y=3\cos ( 2x)\)
  4. (d) \(y=-3\sin ( 2x)\)
  5. (e) \(y=-2\cos\! \left( \dfrac{\pi }{2}x\right)\)
  6. (f) \(y=-2\sin\! \left( \dfrac{1}{2}x\right)\)

(f)

(a)

(d)

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

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

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

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

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

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

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

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

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

\(\frac{1}{2}\), \(2\)

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

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

\(3\), \(2\pi\)

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

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

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

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

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

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

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

\(f(x)=\frac{1}{2}\cos(2x)\)

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

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

\(f(x) = -\sin\left(\frac{3}{2}x\right)\)

\(f(x) = 1-\cos\left(\frac{4\pi}{3}x\right)\)

\(f(x) = \cot x\)

\(f(x) = \tan\left(x-\frac{\pi}{2}\right)\)