cosecant, secant and cotangent
If you flip each of the three basic ratios upside down, you get three more functions: cosecant, secant, and cotangent. They are not new ideas so much as convenient shorthands — names for one-over-sine, one-over-cosine, and one-over-tangent — that make certain formulas tidier and certain calculations shorter.
Precisely, csc theta = 1 / sin theta, sec theta = 1 / cos theta, and cot theta = 1 / tan theta = cos theta / sin theta. A memory aid: the function whose name starts with 'co' is NOT always the reciprocal of the one without 'co'. The pairing is crossed — secant pairs with cosine (sec = 1/cos) and cosecant pairs with sine (csc = 1/sin). In a right triangle this means csc theta = hypotenuse / opposite, sec theta = hypotenuse / adjacent, and cot theta = adjacent / opposite. So in the 3-4-5 triangle with opposite 3, adjacent 4, hypotenuse 5: csc theta = 5/3, sec theta = 5/4, cot theta = 4/3.
These appear most often when integrating or differentiating in calculus and when stating identities compactly — for instance the Pythagorean relatives 1 + tan^2 theta = sec^2 theta and 1 + cot^2 theta = csc^2 theta. On a basic level you can always avoid them by writing reciprocals of sin, cos, and tan; many calculators have no csc, sec, or cot button for exactly that reason.
If sin theta = 0.6 then csc theta = 1 / 0.6 ≈ 1.667. If cos theta = 0.8 then sec theta = 1 / 0.8 = 1.25. And cot theta = cos theta / sin theta = 0.8 / 0.6 ≈ 1.333.
Each reciprocal ratio is just one divided by its partner — no triangle needed once you know sin, cos, tan.
The crossed pairing (sec with cos, csc with sin) trips up almost everyone at first; remember that the third letter of the long name tells you the partner: cosecant -> sine, secant -> cosine.