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Precalculus : Determine the domain of a trigonometric function

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Example Questions

Example Question #1 : Trigonometric Functions

Which of the following is the correct domain of $cot(\theta)+3$ , where represents an integer?

Possible Answers:

$\textup{Only Every: }\pi k$

$\textup{Everywhere except: }\frac{\pi}{2} k$

$\textup{Only Every: }2\pi k$

$\textup{Everywhere except: }\pi k$

$\textup{Everywhere except: }2\pi k$

Correct answer:

$\textup{Everywhere except: }\pi k$

Explanation:

The cotangent graph only has a period of $\pi$ intervals and is most similar to the tangent graph. The domain of cotangent exists everywhere except every $\pi k$ value since an asymptote exists at those values in the domain.

The y-intercept of 3 shifts the cotangent graph up by three units, so this does not affect the domain.

Therefore, the graph exists everywhere except $\pi k$ , where is an integer.

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Example Question #2 : Trigonometric Functions

Please choose the best answer from the following choices.

What is the domain of the following function?

y=sin(x)

Possible Answers:

$\left ( 1,1\right )$

$\left [ 1,1\right ]$

$\left ( -\infty ,\infty \right )$

$\left [ -\infty ,\infty \right ]$

Correct answer:

$\left ( -\infty ,\infty \right )$

Explanation:

All x values make the function work. Thus, making the domain $\left ( -\infty ,\infty \right )$ . They're parentheses instead of brackets because parentheses are used when you can't actually use the specific value next to it. It is impossible to use infinity which makes parentheses appropriate. Brackets are used when you CAN use the specific value next to it.

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Example Question #1 : Trigonometric Functions

Please choose the best answer from the following choices.

What is the domain of the following function:

$y=\cos (x)$

Possible Answers:

$\left ( 0,1\right )$

$\left [ 0,1\right ]$

$\left ( -\infty ,\infty \right )$

$\left [ -\infty ,\infty \right ]$

Correct answer:

$\left ( -\infty ,\infty \right )$

Explanation:

All x values work for the function. Thus, making the domain all real numbers. Parentheses are required because you can never actually use the number infinity.

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Example Question #4 : Trigonometric Functions

Please choose the best answer from the following choices.

What is the domain of $y=\tan (x)$ .

Possible Answers:

$\left ( -\infty ,\infty \right )$

$\left [ -\pi /2,\pi /2\right ]$

$\left \{ x\mid x\neq \pi /2+k\pi \right \}$

$\left ( -\pi /2,\pi /2\right )$

Correct answer:

$\left \{ x\mid x\neq \pi /2+k\pi \right \}$

Explanation:

If you look at a graph of the function, you can see that every curve has a vertical asymptote that repeats every $\pi$ radians in the positive and negative x-direction, starting at $-\pi/2$ radians. Also, the curve has a length that stretches $\pi$ radians which makes the domain $\left \{ x\mid x\neq \pi /2+k\pi \right \}$ .

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Example Question #4 : Determine The Domain Of A Trigonometric Function

What is the restriction of the domain of the function given by:

$y=3sec(3(x-\frac{\pi}{6}))$

For all the answer choices below, is any integer.

Possible Answers:

$x\neq \frac{\pi}{6}n$

$x\neq \frac{2\pi}{3}+\pi n$

$x\neq \frac{\pi}{3}n$

$x\neq \frac{\pi}{3}+\pi n$

Correct answer:

$x\neq \frac{\pi}{3}n$

Explanation:

y=sec(x) has restrictions on its domain such that

$x\neq \frac{\pi}{2}+\pi n$ , where is any integer.

To determine the domain for $y=3sec(3(x-\frac{\pi}{6}))$ ,

we equate the terms within the secant function and set them equal to the original domain restriction.

$3(x-\frac{\pi}{6})\neq \frac{\pi}{2}+\pi n$

Solving for ,

$x-\frac{\pi}{6}\neq\frac{\pi}{6}+\frac{\pi}{3}n$

$x\neq\frac{\pi}{3}+\frac{\pi}{3}n=\frac{\pi}{3}n$

The new domain restriction is:

$x\neq \frac{\pi}{3}n$ where is an integer

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Precalculus : Determine the domain of a trigonometric function

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Trigonometric Functions

Example Question #2 : Trigonometric Functions

Example Question #1 : Trigonometric Functions

Example Question #4 : Trigonometric Functions

Example Question #4 : Determine The Domain Of A Trigonometric Function

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