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ACT Math : How to find the domain of the tangent

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Example Question #1 : How To Find The Domain Of The Tangent

What is domain of the function $\tan(\theta)$ from the interval $[-\pi,\pi]$ ?

Possible Answers:

$\left[-\pi, \frac{-\pi}{2}\right]\cup \left[\frac{\pi}{2},\pi\right]$

$\left[-\pi, \frac{-\pi}{2}\right)\cup \left(\frac{-\pi}{2},\frac{\pi}{2}\right) \cup \left(\frac{\pi}{2},\pi\right]$

$\left(-\pi, \frac{-\pi}{2}\right)\cup \left(\frac{\pi}{2},\pi\right)$

$\left[-\pi, \frac{-\pi}{2}\right)\cup \left(\frac{\pi}{2},\pi\right]$

$\left[-\pi, \frac{-\pi}{2}\right]\cup\left[\frac{-\pi}{2},\frac{\pi}{2}\right] \cup\left[\frac{\pi}{2},\pi\right]$

Correct answer:

$\left[-\pi, \frac{-\pi}{2}\right)\cup \left(\frac{-\pi}{2},\frac{\pi}{2}\right) \cup \left(\frac{\pi}{2},\pi\right]$

Explanation:

Rewrite the tangent function in terms of cosine and sine.

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Since the denominator cannot be zero, evaluate all values of theta where $\cos(\theta)=0$ on the interval $[-\pi,\pi]$ .

$\theta=-\frac{\pi}{2}, \frac{\pi}{2}$

These values of theta are asymptotes and will not exist on the tangent curve. They will not be included in the domain and parentheses will be used in the interval notation.

The correct solution is $\left[-\pi, \frac{-\pi}{2}\right)\cup \left(\frac{-\pi}{2},\frac{\pi}{2}\right) \cup \left(\frac{\pi}{2},\pi\right]$ .

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Example Question #2 : How To Find The Domain Of The Tangent

Where does the domain NOT exist for $y=2tan(x-\frac{\pi}{2})+3$ ?

Possible Answers:

$x=\pi+\pi k \textup{, for k any integer}$

$x=3\pi+\frac{\pi}{2} k \textup{, for k any integer}$

$x=\pi+2\pi k \textup{, for k any integer}$

$x=2\pi+2\pi k \textup{, for k any integer}$

$x=2\pi+\frac{\pi}{2} k \textup{, for k any integer}$

Correct answer:

$x=\pi+\pi k \textup{, for k any integer}$

Explanation:

The domain for the parent function of tangent does not exist for:

$x=\frac{\pi}{2}+\pi k \textup{, for k any integer.}$

The amplitude and the vertical shift will not affect the domain or the period of the graph.

The $x-\frac{\pi}{2}$ tells us that the graph will shift right $\frac{\pi}{2}$ units.

Therefore, the asymptotes will be located at:

$x=\frac{\pi}{2}+\pi k+ \frac{\pi}{2} \textup{, for k any integer.}$

The locations of the asymptotes are:

$x=\pi+\pi k \textup{, for k any integer}$

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Example Question #3 : How To Find The Domain Of The Tangent

Find the domain of $y=tan(x-\frac{3\pi}{4})$ . Assume is for all real numbers.

Possible Answers:

$x=\frac{3\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \frac{\pi}{2} k$

$\textup{Everywhere except: }x=\frac{3\pi}{4}\pm \pi k$

$x=\frac{5\pi}{4}\pm \pi k$

Correct answer:

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \pi k$

Explanation:

The domain of tan(x) does not exist at $\frac{\pi}{2}\pm \pi k$ , for is an integer.

The ends of every period approaches to either positive or negative infinity. Notice that for this problem, the entire graph shifts to the right $\frac{3\pi}{4}$ units. This means that the asymptotes would also shift right by the same distance.

The asymptotes will exist at:

$x=\frac{\pi}{2}\pm \pi k + \frac{3\pi}{4} = \frac{5\pi}{4}\pm \pi k$

Therefore, the domain of $y=tan(x-\frac{3\pi}{4})$ will exist anywhere EXCEPT:

$\frac{5\pi}{4}\pm \pi k$

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ACT Math : How to find the domain of the tangent

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Find The Domain Of The Tangent

Example Question #2 : How To Find The Domain Of The Tangent

Example Question #3 : How To Find The Domain Of The Tangent

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