Secant, Cosecant, Cotangent

Each of the three has a reciprocal with a name: $\sec\left(\theta\right) = \frac{1}{\cos\left(\theta\right)}$, $\csc\left(\theta\right) = \frac{1}{\sin\left(\theta\right)}$ and $\cot\left(\theta\right) = \frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}$. On the unit circle that is 1/x, 1/y and x/y. The names pair crosswise: secant goes with cosine, cosecant with sine. Each pair has exactly one "co".

To find one at any angle, find its partner, then flip. $\sin\left(150^{\circ}\right) = \frac{1}{2}$, so $\csc\left(150^{\circ}\right) = 2$. Flipping never changes a sign, so each reciprocal has its partner's sign in every quadrant.

Flipping 0 is impossible: secant is undefined where cosine is 0 (90°, 270°), cosecant and cotangent where sine is 0 (0°, 180°).