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ACT Math : Trigonometry

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Example Questions

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

What is the period of the following tangent function?

f(x)=3tan(5x)+4

Possible Answers:

$\frac{2\pi}{5}$

$\frac{\pi}{5}$

$3\pi$

$\frac{\pi}{4}$

Correct answer:

$\frac{\pi}{5}$

Explanation:

The period of the tangent function defined in its standard form f(x)=tan(x) has a period of $\pi$ . When you multiply the argument of the trigonometric function by a constant, you shorten its period of repetition. (Think of it like this: You pass through more iterations for each value that you use.) If you have f(x)=tan(5x) , this has one fifth of the period of the standard tangent function. In the equation given, none of the other details matter regarding the period. They alter other aspects of the equation (its "width," its location, etc.). The period is altered only by the parameter. Thus, the period of this function is $\frac{1}{5}$ of $\pi$ , or $\frac{\pi}{5}$ .

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

What is the period of the following trigonometric equation:
$h(t) = 4\tan(-3t)$

Possible Answers:

$\frac{\pi}{2}$

$-\frac{\pi}{3}$

$2\pi$

$4\pi$

$\frac{\pi}{3}$

Correct answer:

$\frac{\pi}{3}$

Explanation:

For tangent and cotangent the period is given by the formula:

$\omega = \frac{\pi}{\left | b\right |}$ where comes from $h(t)= a\tan(bt)$ .

Thus we see from our equation b=-3 and so
$\omega = \frac{\pi}{\left | -3\right |}=\frac{\pi}{3}$ .

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

What is the period of the trigonometric function given by:
$h(t) = -4\tan(-3t)$ ?

Possible Answers:

$\frac{\pi}{4}$

$-\frac{\pi}{3}$

$\frac{2\pi}{3}$

$\frac{\pi}{3}$

$-\frac{2\pi}{3}$

Correct answer:

$\frac{\pi}{3}$

Explanation:

To find the period of a tangent funciton use the following formula:

$\omega = \frac{\pi}{\left | b\right |}$ where comes from $f(t) = a\tan(bt)$ .

thus we get that b = -3 so $\omega = \frac{\pi}{\left|3\right|} = \frac{\pi}{3}$

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

What is the period of the following trigonometric function:

$m(t) = -3\cot(-2t)$

Possible Answers:

$\omega = \frac{\pi}{3}$

$\omega = -\frac{\pi}{2}$

$\omega = -\pi$

$\omega = \frac{\pi}{2}$

$\omega = \pi$

Correct answer:

$\omega = \frac{\pi}{2}$

Explanation:

To find the period of a tangent or cotangent function use the following formula:

$\omega = \frac{\pi}{\left| b\right|}$

from the general tirogonometric formula:

$m(t) = a\cot(bt)$

Since we have,

$m(t) = -3\cot(-2t)$

we have

|b|=|-2|=2 .

Thus we get that

$\omega = \frac{\pi}{2}$

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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:

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

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

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

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

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

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:

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

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

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

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

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

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:

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

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

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

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

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

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

Find the range of: $y= 2\tan(\theta)+15$

Possible Answers:

$(15,\infty)$

$[15,\infty)$

$[7.5,\infty)$

${}(-\infty,\infty)$

$(7.5,\infty)$

Correct answer:

${}(-\infty,\infty)$

Explanation:

The function $y= 2\tan(\theta)+15$ is related to $y=\tan(\theta)$ . The range of the tangent function is ${}(-\infty,\infty)$ .

The range of $y= 2\tan(\theta)+15$ is unaffected by the amplitude and the y-intercept. Therefore, the answer is ${}(-\infty,\infty)$ .

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Example Question #31 : Tangent

What is the range of the trigonometric fuction defined by:
$f(t) = -2 \tan(3t)$ ?

Possible Answers:

$\mathbb{N}$

$\mathbb{R}$

$\mathbb{Z}$

[-2,2]

[-1,1]

Correct answer:

$\mathbb{R}$

Explanation:

For tangent and cotangent functions, the range is always all real numbers.

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Example Question #32 : Tangent

What is the range of the given trigonometric function:

$n(t) = 4\cot(2t)$

Possible Answers:

[0,2]

[-4,4]

$(-\infty, \infty)$

[-2,2]

[-4,0]

Correct answer:

$(-\infty, \infty)$

Explanation:

The range of a function is every value that the funciton's results take. For tangent and cotangent, the function spans from $(-\infty, \infty)$ and so the range is:

$(-\infty, \infty)$

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ACT Math : Trigonometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

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

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

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

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

Example Question #31 : Tangent

Example Question #32 : Tangent

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