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According to Sine - Wolfram Mathworld, Sine is defined as $$\sin(\theta) = \frac{\operatorname{opposite} (a)}{\operatorname{hypotenuse} (b)}$$ enter image description here

From that definition, I don't understand why the sine value in the unit circle in the 3rd quadrant is negative; after all, $a$ and $b$ should both be negative there and a positive quotient follows from this:

enter image description here

People have already explained to me that b is always positive in this case, since it represents the radius and not a vector like a or c (not drawn in here now).

Which leads me to the final question, how is one supposed to actually know that, based on the above definition?

I mean it must be explicitly mentioned somewhere that $b$ in this case represents a length and $a$, $c$ vectors that can take on both positive and negative values; the only thing I see on many pages is that a coordinate system is drawn into the unit circle and hence my obviously wrong assumption.

iwab
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  • What is the sign of $a$ when the distance $b$ is rotated counterclockwise from the $x$-axis into the third quadrant? It's negative. Way overthinking this point. – Nij Aug 15 '23 at 22:48
  • @Nij Well, that is the point. a is negative (but b should be negative too, leading to a positive quotient) – iwab Aug 15 '23 at 22:55
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    In this context, $b$ represents the radius of a circle. $b$ is positive. When $b$ is negative, it would go backwards through the circle center, but then you wouldn't be measuring $200^\circ$ or whatever you're taking the sine of. – PrincessEev Aug 15 '23 at 22:57
  • In general, you can think of trigonometric ratios as ratios of $x$- and $y$-coordinates on a circle of radius $r$. You can see a visual here along with other answers. – Accelerator Aug 15 '23 at 23:01
  • @PrincessEev I understand, just wondering how I should derive from the definition of mathworld, for example, without help that b is always positive, because it doesn't really emphasise that everything except b is a vector. – iwab Aug 15 '23 at 23:02
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    @Leonard It's overkill to call $a$ a "vector." It's just a number. Numbers are vectors, but not really. It's the $y$ coordinate of the point on the circle. It can be positive or negative. $b$ is a number too, but it's a length, so it's never negative. You can't proceed without knowing that $b$ is not negative, so it's pointless to try to imagine a situation where you could deal with this without knowing that. – Matt Samuel Aug 15 '23 at 23:15
  • @MattSamuel So if I hadn't asked anyone and didn't know, would I have come up with it myself at some point because many other things wouldn't have worked with the assumption that b could be positive and negative depending on the orientation? – iwab Aug 15 '23 at 23:25
  • I don't know. That would depend on you. – Matt Samuel Aug 15 '23 at 23:26
  • @MattSamuel well, just based on correct, logical reasoning – iwab Aug 15 '23 at 23:27
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    @Leonard You identified that something doesn't make sense. You asked a question on Math.SE about it, which is a reasonable thing to do. It was clarified for you. It doesn't seem worthwhile to dwell on what would've happened if you didn't do that. Just my perspective. – Matt Samuel Aug 16 '23 at 00:13
  • @Leonard: This old answer of mine may be helpful. – Blue Aug 16 '23 at 13:15

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From the site you quote (bold mine):

"The common schoolbook definition of the sine of an angle theta in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths ..."

An angle $\theta>\pi$ cannot be seen as an angle in a right triangle.

So you need an extended definion of the trig functions that includes all angles. I suggest you use the following:

Let $C$ the circle centered at $O=(0,0)$ and of radius $1$. Let $P\in C$ a point such that the segment $OP$ forms an angle $\theta$ with the $x$-axis. Then $$ P=(\cos\theta,\sin\theta). $$

Andrea Mori
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  • Thanks :-) After your definition no wrong interpretation area remains more on. Maybe I just transferred the triangle from the triangle definition to the unit circle definition too fast. For me it did not seem logical to assume that in the same geometrical object (here the right triangle in the unit circle) 2 parts follow the coordinate system, while 1 part follows only the magnitude. Do you have a good resource for trigonometry that eliminates such confusion or did I just reason wrong? – iwab Aug 16 '23 at 15:33