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$\sin \theta= -\frac{3}{7}$.

Find $\csc\theta$.

I don't get it, is $x$ or $r$ the negative?

There just doesn't seem to be enough information given.

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$r=\sqrt{x^2+y^2}\ge 0$. – Américo Tavares Jun 1 '11 at 20:29
Typically $x$ and $y$ can each be positive or negative (and one of them can be $0$), while $r$ is always positive. – Henry Jun 1 '11 at 22:09
up vote 2 down vote accepted

$\sin$ is a function which takes any real argument, so in general you don't really need to think of it in terms of triangles.

But let's say that you want to think of it exclusively in terms of triangles, so that $\sin(\theta)$ is defined as follows: draw a right triangle with angle $\theta$; then $\sin(\theta)$ equals the length of the opposite side divided by the length of the hypothenuse; if we place the adjacent side on the positive $x$-axis, then length is taken to be "positive" if it goes to the right or up, and negative if it goes left or down. Then, it doesn't matter if you are considering the opposite side or the radius to be "negative", what matters is:

  1. The value of $\sin\theta$; and
  2. The relationship between $\sin\theta$ and $\csc\theta$.

Remember that, as long as $\sin\theta\neq 0$, then you have that $$\csc\theta = \frac{\text{hypothenuse}}{\text{opposite}} = \frac{1}{\quad\frac{\text{opposite}}{\text{hypothenuse}}\quad} = \frac{1}{\sin\theta}.$$

So, since you know what $\sin\theta$ is, how much is $\csc\theta$?

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I suppose saying $\sin$ takes any real argument and then talking about hypotenuse might confuse some readers :-) – Aryabhata Jun 1 '11 at 20:22
@Aryabhatta: I suppose you are definitely right... – Arturo Magidin Jun 1 '11 at 20:27
I don't understand a thing said in this thread so far, but I did just realize that sin is y/r and csc is r/y so it is -7\3 osm – Adam Jun 1 '11 at 20:37
@Adam: If you start at "Remember that, as long as...", isn't that the same thing you just wrote? Note that $r/y$ is the reciprocal of $y/r$. Sorry that this was all completely unintelligible. – Arturo Magidin Jun 1 '11 at 20:38
@Adam: Please don't feel pressured to "accept" an answer. This is particularly true when you are having trouble understanding the answer, as you do here. I would suggest that you "unaccept" this answer and wait; perhaps someone else will be able to explain things in a way that you can understand more easily. – Arturo Magidin Jun 1 '11 at 20:49

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