I'm trying to wrap my head around sine, cosine, and tangent. I'm aware that they're commonly defined in high schools as ratios of the various parts of triangles set in the unit circle, but that's not particularly intuitive.
Can the trigonometric functions be expressed, explained, or proven in terms of something more intuitive, say, arithmetic? Is there a way to work out sine, cosine, and tangent on paper, without the aid of a calculator or computer?
How were they first proven or discovered? What proof might have been worked through to reach the trigonometry we have today?
Edit:
I discussed this question with a friend today. I think ultimately, what I'm looking for, is a way to work out the sine, cosine, or tangent of a given angle using paper and pencil.