Cosecant, Secant, Cotangent

Turn each of the three ratios upside down and you get three more: the same two sides, the other way up.

$$\csc\theta = \frac{1}{\sin\theta} = \frac{\text{hypotenuse}}{\text{opposite}}$$

$$\sec\theta = \frac{1}{\cos\theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$$

$$\cot\theta = \frac{1}{\tan\theta} = \frac{\text{adjacent}}{\text{opposite}}$$

In an 8, 15, 17 triangle with $\theta$ opposite the 8: sin θ = 8/17, so $\csc\left(\theta\right) = \frac{17}{8}$; cos θ = 15/17, so sec θ = 17/15; tan θ = 8/15, so cot θ = 15/8. The pairing is the one trap: the co names cross over, so cosecant goes with sine and secant with cosine.

An equation with a reciprocal is solved by flipping it. $\csc\left(\theta\right) = \frac{17}{8}$ is the same as $\sin\left(\theta\right) = \frac{8}{17}$, and the board's chip Use arccsc: every angle in the domain does the flip and the arcsin in one step.