The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. The word “sine” comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.
I asked a guy I know from Jordan about the word. The non-trigonometry root of the word “sine” is akin to sinus. It is “pocket” or “cave.” So I asked for the word, in Arabic, for “pocket” and he said what I would write as “jaib” or “jahib.” So, I then asked him the word for “sine” and he said “jiba.” In Arabic the pronunciations of these two words are so similar. My guess is that “jiba” became heard as “jaib” and thus the mistake. Tangent and secant function names are obvious, pocket is not.
I have 2 interactive geogebra graphics below. The first one is less busy and gets the basic three functions of trigonometry represented. The second is all six functions. If you “grab” point B and drag it around the circle, you can see the pieces move and how they relate to the angle THETA. You can also click the play button in the bottom left corner.