Why do sine and cosine have values at 90° and 0°?

A premise says trigonometric function can only be applied in a right triangle. There seems to be a logical contradiction. How can you have a right triangle with two 90° or with 0° ?

Or put another way, as θ is zero, we have a line. As it goes from 0° to 90°, the cathetus opposite to θ get’s bigger and bigger, but it cannot reach 90° since that kind of triangle does not exist.

The same applies to cosine. Let θ be 0°. Is that a (right) triangle? Nope. Yet cosine still equals 1 when θ equals 0° and also cosine equals 0 when θ equals 90°. When θ reaches 90° the cathetus perpendicular to cosine cathetus becomes the hypotenuse.

To finalize, I see two problems in these values.
-How can a triangle have two 90° angles?
-When θ reaches 90°, the cathetus opposite to it becomes a hypotenuse, therefore we actually calculate cosine by dividing two cathetus, which is not the definition of cosine or but otherwise, when θ is 90° we calculate sine by dividing hypotenuse with catethus, which again, is not the way sine is calculated.