A Ladder and a Secant

A reciprocal ratio often states a comparison directly. A rope that is twice as long as its peg is far from the mast has hypotenuse over adjacent equal to 2, which is sec θ = 2, so cos θ = 1/2 and the rope stands at 60° to the ground.

Off the special values the flip works the same way. If a cable is 1.5 times as long as the distance from its anchor to the pole, $\sec\left(\theta\right) = 1.5$, so cos θ = 1/1.5 = 2/3 and θ = arccos(2/3), about 48.19°. On the board, Use arcsec: every angle in the domain writes θ = arcsec(1.5) and the round chip finishes it.