The flat end and the upright end

Press a ladder flat to the ground. Stand it straight up against the wall. What do the ratios do?

The two end angles

Let A grow from 0° to 90°. Sin A grows from 0 to 1. Cos A falls from 1 to 0. Tan A grows from 0 without end.

No right triangle has an acute angle A of 0° or 90°. So we use the value each ratio gets close to.

Keep the hypotenuse fixed. Let A shrink toward 0°. The opposite side shrinks toward 0. The adjacent side grows to match the hypotenuse. So sin 0° = 0 ÷ hypotenuse = 0. And cos 0° = hypotenuse ÷ hypotenuse = 1. Then tan 0° = 0 ÷ adjacent = 0.

Now let A grow toward 90°. The adjacent side shrinks toward 0. The opposite side grows to match the hypotenuse. So sin 90° = hypotenuse ÷ hypotenuse = 1. And cos 90° = 0 ÷ hypotenuse = 0.

Tan A is opposite ÷ adjacent. Near 90° the adjacent side is tiny. So tan A grows without end. At 90° the adjacent side would be 0. Dividing by 0 has no answer. So tan 90° is not defined.

Rule: sin 0° = 0, cos 0° = 1, tan 0° = 0. sin 90° = 1, cos 90° = 0. tan 90° is not defined.