[Math] The sine/cosine/tangent of 0, 180 and 360 degree angles

geometrytrigonometry

If the ratios sine, cosine and tangent are applicable to right angled triangles, then how can they be applicable for the angles of 0, 180 and 360 degrees?
They don't seem to have a right angle, since there more like a flat line.

For example, the cosine of 0° is 1, but doesn't this contradict the required 90% straight angle to make the rule valid (?).
The formula adjacent side / hypothenuse would ultimately "appear" to result into 1 if they would fall together, but what is the proof they actually get there in the scope of trigonometry?
Why can't there be assumed an "infinite approximation" instead, without it ever getting to 1?

Best Answer

The $\sin$ function has a definition that goes beyond simply right angled triangles.

The most useful definition for high-school level math is this:

  • $\sin x$ is defined as the $y$-coordinate of a point $P$ on the unit circle for which the length of the arc between $(1,0)$ and $P$ is equal to $x$
  • $\cos x$ is the $x$-coordinate of the same point.
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