1
$\begingroup$

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

Find $\csc\theta$.

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

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

  • 2
    $r=\sqrt{x^2+y^2}\ge 0$.2011-06-01
  • 0
    Typically $x$ and $y$ can each be positive or negative (and one of them can be $0$), while $r$ is always positive.2011-06-01

1 Answers 1

2

$\sin$ 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 $\csc\theta$?

  • 0
    I suppose saying $\sin$ takes any real argument and then talking about hypotenuse might confuse some readers :-)2011-06-01
  • 0
    @Aryabhatta: I suppose you are definitely right...2011-06-01
  • 1
    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 osm2011-06-01
  • 0
    @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.2011-06-01
  • 0
    Sorry I don't understand what any of that is, I will have to look up what a hypthohenuse is and I am not familiar with the term opposite in this context.2011-06-01
  • 0
    @Adam: The hypothenuse of a right triangle is the longest of its three sides. The opposite side of an angle is the side that is in the "opposite site" of the triangle (the side that does not touch the angle). Your "r" is the length of the hypothenuse; your "y" is the length of the opposite side.2011-06-01
  • 1
    @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.2011-06-01