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For the unit circle on a X-Y plane, is there a name for the Angle a radial line makes with the positive X-axis? The closest name that I can get from Wikipedia is a 'Central Angle' ( http://en.wikipedia.org/wiki/Central_angle ) but for this definition the base reference line is not required to be the positive X-axis. I am not sure there is even a name for this angle.

The other way of putting this question is... is there a special name for the angle theta used in Sin(theta), Cos(theta), etc..

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  • $\begingroup$ I am not sure what exactly, you are looking for, but the terms "base angle" comes into my mind. There's also angular offset, which is simply the angle between two arbitrary lines. $\endgroup$ – Yiyuan Lee Mar 4 '14 at 9:12
  • $\begingroup$ From Wikipedia: "A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B". I want to know if there is a name for a 'central angle' with the condition that either A or B should be the positive X-axis. $\endgroup$ – kums Mar 4 '14 at 9:26
  • $\begingroup$ Another question is.. instead of using 'theta' for the angle between two arbitrary lines, is there a notation like theta<sub>X</sub> to imply the angle is between the positive X-axis and another line? $\endgroup$ – kums Mar 4 '14 at 9:45
  • $\begingroup$ I am still unable to find a name for the angle you are looking for! We can use other symbols such as $\phi,\psi$ to represent angles. Perhaps you could define them to be whatever you need in your workings. $\endgroup$ – Yiyuan Lee Mar 4 '14 at 9:46
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An angle measured relative to the positive x-axis in the counter-clockwise (anti-clockwise) direction is said to be in standard position. The side along the positive x-axis is called the initial side of the angle. The other side is called the terminal side of the angle. I do not know a special name for the angle other than saying it is in standard position.

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Preliminary remark: The name of the angle theta used in sin(theta) is theta. Note that we are free to choose the names of (or variables for) individual elements in our figures and structures. Certain famous mathematical objects, like ${\mathbb R}$ or $S^2$, have well-established names which should not be used for other purposes. But harmless letters like $\alpha$, $\theta$, $x$, $\phi$, etc. can be used for most anything.

One of the most important functions in analysis is the function $$\arg:\quad \dot{\mathbb R}^2\to{\mathbb R}/(2\pi),\quad{\rm resp.},\quad \dot{\mathbb C}\to{\mathbb R}/(2\pi),$$ where $\dot{\mathbb R}^2:={\mathbb R}^2\setminus\{(0,0)\}$, and similarly for $\dot{\mathbb C}$. This function is written as $$\arg(x,y), \quad \arg(x+iy),\quad{\rm or}\quad \arg(z),$$ depending on context. It gives the angle you are talking about "up to multiples of $2\pi$". If you remove the negative $x$-axis (resp., negative real axis) from $\dot{\mathbb R}^2$ (resp., from $\dot{\mathbb C}$) you can single out the principal value of the argument, denoted by ${\rm Arg}(x,y)$, which is then a well defined continuous real-valued function on this restricted domain, taking values in $\ ]-\pi,\pi[\ $. One has $${\rm Arg}(x,y)=\arctan{y\over x}\qquad(x>0)$$ and similar formulas in other half planes.

Even though the values of $\arg$ are not "ordinary real numbers" the gradient of $\arg$ is a well defined vector field in $\dot{\mathbb R}^2$, and is given by $$\nabla\arg(x,y)=\left({-y\over x^2+y^2},\>{x\over x^2+y^2}\right)\qquad\bigl((x,y)\ne(0,0)\bigr)\ .$$

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  • $\begingroup$ I'm sorry, but what does the point over the letter $\mathbb R$ stand for? $\endgroup$ – Ant Sep 10 '14 at 13:08
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polar coordinate angle, ( the other part is polar coordinate radius).

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Call it the argument of the trigonometric function you are using.

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