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Thread: Reciprocal Trigonometric Functions

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    Reciprocal Trigonometric Functions

    Hi,

    I hope someone can help. I'm trying to understand the domain for the reciprocal trigonometric functions. As follows:

    For secant function:
    Reciprocal Trigonometric Functions-screen-shot-2017-05-16-6.20.56-pm.png

    For cosecant function:
    Reciprocal Trigonometric Functions-screen-shot-2017-05-16-6.20.48-pm.png

    For cotangent function:
    // same as secant function's domain

    I know that the domain must exclude values that make the function undefined, but I don't see how that is being communicated. Can someone please help me decipher the domain annotation above?

    Sincerely,
    Olivia
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    Re: Reciprocal Trigonometric Functions

    secant is the reciprocal of cosine, so wherever $\cos{x} = 0$, $\sec{x}$ is undefined.

    $\cos{x} = 0$ at odd-integer multiples of $\dfrac{\pi}{2}$ ... so, the domain of $y=\sec{x}$ is all reals except $\color{red}{(2k-1)}\dfrac{\pi}{2} \, ; \, k \in \mathbb{Z}$

    cosecant is the reciprocal of sine, so wherever $\sin{x} = 0$, $\csc{x}$ is undefined.

    $\sin{x} = 0$ at integer multiples of $\pi$ ... so what is the domain of $y=\csc{x}$?
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    Re: Reciprocal Trigonometric Functions

    Quote Originally Posted by otownsend View Post
    Hi,

    I hope someone can help. I'm trying to understand the domain for the reciprocal trigonometric functions. As follows:

    For secant function:
    Click image for larger version. 

Name:	Screen Shot 2017-05-16 at 6.20.56 PM.png 
Views:	1 
Size:	11.6 KB 
ID:	37602

    For cosecant function:
    Click image for larger version. 

Name:	Screen Shot 2017-05-16 at 6.20.48 PM.png 
Views:	1 
Size:	9.1 KB 
ID:	37603

    For cotangent function:
    // same as secant function's domain

    I know that the domain must exclude values that make the function undefined, but I don't see how that is being communicated. Can someone please help me decipher the domain annotation above?

    Sincerely,
    Olivia
    This is notation for defining a set. $\{1\}$ is the set containing the number 1. $\{x\in \mathbb{R}\}$ is the set of all real numbers. This notation uses the curly braces to denote a set, then it tells you that any element $x$ must be an element of the set of real numbers. In other words, $\{x \in \mathbb{R}\} = \mathbb{R}$.

    Now, your book seems to be using $\mathbb{I}$ as the set of integers. Typically, in mathematics, the set of integers is written $\mathbb{Z}$ because the German word for countable numbers is Zahlen.

    Using your book's notation, the set of integers can be written as $\mathbb{I}$ or $\{n \in \mathbb{I} \}$. Notice that I changed the variable from $x$ to $n$. The choice of variable is mostly arbitrary. We could, in theory, use any variable inside the curly braces to indicate an arbitrary element of the set. That means that $\{n \in \mathbb{I}\} = \{ \zeta \in \mathbb{I}\}$ because again, the choice of variable I use is not important. What is important is that I say that the variable (or generic element of my set) must be a member of the set of all integers.

    Next, what if we wanted to represent the set of all odd integers. In this case, we would use the vertical line you see above. That indicates that there are conditions placed on elements in the set. For example: $\{2n+1 | n\in \mathbb{I} \}$ would be read that any element of the set takes the form $2n+1$ where $n$ is any integer. For example, $2(0)+1 = 1$ is in the set. $2(1)+1 = 3$ is in the set. $2(-1)+1 = -1$ is in the set. The set that I just wrote is just one of many ways to represent the set of all odd integers. Above, they give $\{2n-1|n\in \mathbb{I}\}$.

    Now, finally, we can explain what the notation you are curious about means. For the first one, it means $\theta$ is a real number, but $\theta$ is not equal to $(2n-1)\dfrac{\pi}{2}$ where $n$ is an integer. As skeeter showed you, these excluded values for $\theta$ are all of the values for $\theta$ where $\cos \theta$ will be zero causing the reciprocal to be undefined.

    For the cosecant function, it reads, $\theta$ is a real number, but $\theta$ does not equal $n\pi$ for any integer value of $n$. Again, skeeter explained why we must exclude these values.
    Last edited by SlipEternal; May 16th 2017 at 03:36 PM.
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    Re: Reciprocal Trigonometric Functions

    Thanks for responding! But how does pi/2 represent an "odd integer multiple"?
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    Re: Reciprocal Trigonometric Functions

    Quote Originally Posted by otownsend View Post
    Thanks for responding! But how does pi/2 represent an "odd integer multiple"?
    $2n-1, n\in \mathbb{I}$ represents an odd integer. $(2n-1)\dfrac{\pi}{2}, n\in \mathbb{I}$ represents an odd integer multiple of $\dfrac{\pi}{2}$. I do not understand your question. $\dfrac{\pi}{2}$ does not represent a multiple of anything. It is the thing being multiplied.
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  6. #6
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    Re: Reciprocal Trigonometric Functions

    Never mind, I understand the notation for the secant function now! Sorry for the confusion.
    Thanks from SlipEternal
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