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I was doing a bunch of questions asking if certain inputs for sin would be periodic.

I am not asking for a proof that sin is periodic, I understand that sin is periodic, I am asking why geometrically speaking would a function that takes the angle and outputs the ratios between 2 sides of the triangle would be periodic?

it's obvious why it would be for x + 360, why would it be for 180-x? can't really draw right angle triangles with angles like 135 can we?

I have thought about the 180 - x thing for quite a while with no results, looking it up also got no relevant results

I have always thought about trignometry in terms of right angle triangles, maybe this is wrong.

Blue
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Arsh Dixit
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    If you want to input values outside the interval $(0^\circ,90^\circ)$ into $\sin$, then you need to expand outside your basic opposite-over-hypotenuse right-angled triangle definition. – Arthur Nov 02 '21 at 08:04
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    Saying that $f(\pi- x)=f(x)$ for all $x$ has nothing to do with $f$ being periodic. If you're fine with the fact that $\sin(x+2\pi)=\sin(x)$ for all $x$, then that's all she wrote as far as periodicity of $\sin$ is concerned. –  Nov 02 '21 at 08:04
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    Quoting wikipedia: “The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle are often used..... – ryang Nov 02 '21 at 08:08
  • .... While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between $0$ and $\frac{\pi}{2}$ radian $(90°),$ the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.”

    P.S. The trigonometric functions can be defined using a general circle instead of a unit circle.

     – ryang Nov 02 '21 at 08:08
  • you can think sin(t) as the x-coordinate of the two variable function(sin(t),cos(t)), geometrically, which trances a circle on the interval $[0,2\pi]$ and repeats the same for other values. Also use 2pi radian=360 degrees if you want in terms of degree. – sabeelmsk Nov 02 '21 at 08:19
  • This answer of mine may help explain extending the geometric definitions of sine and cosine to angles beyond the right angle by introducing signs. It focuses on breaching the $90^\circ$ barrier, but the idea can carry you around the circle. This answer illustrates how the trig values of various angles around the unit circle amount to signed distances on "reference triangles". – Blue Nov 02 '21 at 08:42
  • Tangential comment, but the unit circle definition of sine and cosine is very natural if we're interested in describing periodic motion. The simplest nontrivial example of periodic motion is a particle moving along a circle at constant speed. It is the very essence of periodic motion. The coordinates $x(t)$ and $y(t)$ of such a particle are the simplest nontrivial real-valued periodic functions. – littleO Nov 02 '21 at 09:35

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To get a right triangle, consider an obtuse triangle $ABC$ such that $B > 90^\circ$. Here is an illustration.

enter image description here

Extend $\overline{AB}$ to $P$ such that $\overline{AP} \perp \overline{CP}$. We now have a right triangle $APC$ where $\angle APC = 90^\circ$. We also have another right triangle $BPC$ where $\angle BPC = 90^\circ$ and $\angle PBC = 180^\circ - \angle ABC$.

Set $\alpha = \angle CAP$, $\beta = \angle CBP$, and $h = \overline{CP}$. Clearly, $\sin \alpha = h$. We also have $\sin \beta = h$. You can get the rest.

soupless
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