Sign convention in trigonometric functions

Question

So, as I have read and even been taught by my teachers, sign convention in trigonometric functions is based on the location of the respective x and y points denoting the coordinates of a particle going around a circle. Although I am kind of sure that I am right but I still want to confirm this one thing:

When determining the slope of a graph, we encounter both obtuse and acute angles (the angle made by the tangent of the graph at a particular point on it with the x axis). So, is it just a coincidence that tan(a)(the angle made by the tangent) is positive in case of acute angles and negative in case of obtuse angles in both this graphical sense as well as tan(a) for a point on a circle?

Because if a is obtuse(graph), that would mean that the quantity in the y axis is decreasing and if it is increasing, the angle is positive. So, I just want to ask that here tan(a) just happens to be equal to the slope of the graph right? I checked for other trigonometric functions and well sin(a) seems to be negative if the angle is obtuse in the graphical sense but positive in the circular sense.

Edit: I have added a picture for clarity about my definition of graphical and circular sense.

Answer 1

I think I get Where your confusion lies,Though comment made by @Jmoravitz is accurate to point that we should have same language to talk about things,in this case,Definitions matter a lot, Now you, due to some reason I don't follow,Try to differentiate between graphical sense and circular sense, The trigonometric functions for "any angle" are defined using unit circle,This does not mean they might not apply to angles in "graphical sense" that you seem to perceive,Angles are angles,No matter where they are,either associated to unit circle,Or between some random curve's tangent and "x-axis"

Now,trigonometric functions such as $\tan\theta$,Here,happens to be slope of Tangent making angle:$\theta$ with x-axis,If slope of Tangent would have happened to be Some other function,We would have used That Function instead Of $\tan\theta$

Now one could also argue that It is intuitive why tangent function and Slope of tangent relate for positive slopes ,i.e $0\leq\theta\leq90°$,Because before extension of Trigonometric ratios to trigonometric functions, the definition of slope Of tangent and definition of $\tan\theta$ Co-incides ,or precisely the same,

To be clear the extension of definition of trigonometric ratios to functions was not a co-incidence,Mathematicians already knew that they would have to include negative slope in order to again co-incides the tangent function with slope

Answer 2

When a line crosses the $x$ axis, four angles are formed. Two of the angles are acute (and equal magnitude). The other two angles are obtuse (and equal magnitude).

In the unit circle definition of an angle, angles have a direction as well as a magnitude. If you travel counterclockwise around the circle you get a positive angle. If you travel clockwise, the angle is negative.

When we speak of the angle at which a tangent line crosses the axis, we usually mean the acute angle and not the obtuse angle, even though there is always an obtuse angle whenever there is an acute angle.

Moreover, we think of the angle as having a direction like the angles we can make in the unit circle. To make an acute angle above the $x$ axis and below the tangent line, you can rotate a line from the $x$ axis counterclockwise through an acute angle. So the resulting angle value is positive. But to make an acute below the $x$ axis and above the tangent line, you would rotate the line clockwise, so the angle value is negative.

In this way you always get an angle between $-90^\circ$ and $90^\circ$.

Another way to look at this is that we want to compare the direction of the right half of the $x$ axis withe the right half of the tangent line, not the top half of the tangent line. Let the value of $x$ be increasing on both sides of the angle.

If you know the slope of a tangent line and you want to know its angle, the standard method is to find the arc tangent of the slope. Again, this always gives an answer between $-\pi/2$ radians and $\pi/2$ radians, that is, between $-90^\circ$ and $90^\circ$.

If you choose to consider an angle that is different from the angle everyone else is looking at, you may get unusual results.

Another thing you are doing differently is to ask about the sines of angles. The slope of a line at angle $\theta$ is $\tan(\theta)$, not $\sin(\theta)$. But I think that the choice of how to describe the angle between a line and the $x$ axis is the main concern here, because you’re looking at the tangent line with assumptions that will make it hard to understand how other people talk about it.

It is an excellent thing that you have chosen to ask this question, however, because every apparent contradiction we find in standard mathematics is a sign that we missed how other people defined things (or that we made some mistake), and this kind of confusion can make it very difficult to do mathematics. And I don’t know any solution to that difficulty other than asking.