I have a pre calculus test next week, and I have been going over the chapters to gain a deeper understanding; however, I find it difficult, or at least I find the concept of "Angles in Standard Postion" confusing.

The book states "An angle is in standard position if it is drawn in the xy-plane with its vertex at the origin and its initial side on the positive x-axis"

I guess the best question I can ask is when is an angle not in "Standard Position"?

Please keep in mind that I am a math novice.


2 Answers 2


Standard angles demonstration

For an angle to be in standard position, both of the following must be true

  • the initial side (see blue ray above) must lie on the right side of the $x$-axis
  • the vertex (see red dot) must be located on the origin, which is the intersection of the $x$ and $y$ axes
  • $\begingroup$ Great explanation. Thank you so much. $\endgroup$
    – user137452
    Commented Jul 10, 2014 at 22:54
  • $\begingroup$ The point where the terminal side and the unit circle intersect is ($\cos\theta$,$\sin\theta$). $\endgroup$
    – John Joy
    Commented Jul 12, 2014 at 14:10

Stand outside this evening. Point one arm at the north star, another at, say, Sirius. Your two arms define an angle, but unless you regard your chest as "the origin" and the $x$-axis as pointing from your chest to Sirius or Polaris, then the angle is not in standard position. In fact, draw any two rays that start from the same point $P$ in the plane. If $P$ is not the origin, then the angle isn't in standard position. Draw a ray from the origin to $(1,1)$, and another from the origin to $(0, 1)$; that angle's not in standard position.

So why define standard position at all? Because it gives us a basis for comparing things. By translating and rotating, any angle may be moved to the origin, with its first ray along the positive-$x$ axis. If you do that with two different angles, you can compare them, saying that one's larger than the other, etc. Standard position is also a convenient place to define "sine" and "cosine".

By the way, the idea of moving things to the origin to compare them probably seems silly. Why not just measure the angle? In the plane, you can do that. But one of the great puzzle of physics came in the 1920s/1930s when people asked "Are two measurements of the same thing always the same? For instance, if you measure the distance between two goalposts as you stand on the ground between them, and I measure the distance by "instantaneously" noting where each post is, as I travel past them on a train, will we get the same distance? What if the train is moving at nearly the speed of light? Turns out that the answer's not so simple. So while this "standard of measure" idea seems stupid for angles in the plane, it's setting you up for deeper questions about measurement later on.


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