Mathematics

The key insight is to use the Pythagorean identity to build a right triangle from the given cotangent ratio. Since $\cot\phi = k = \frac{\text{adjacent}}{\text{opposite}}$, you can set the opposite side to 1 and adjacent side to $k$, then find the hypotenuse using the Pythagorean theorem. Once you have all three sides, all six ratios follow directly from their definitions.

The key insight is to use the Pythagorean identity to build a right triangle from the given cotangent ratio. Since $\cot\phi = k = \frac{\text{adjacent}}{\text{opposite}}$, you can set the opposite side to 1 and adjacent side to $k$, then find the hypotenuse using the Pythagorean theorem. Once you have all three sides, all six ratios follow directly from their definitions.