Question

If you have a 10 foot ladder and you put it against a wall at the ideal angle how high on the wall will the ladder touch assume a 90 degree angle between the wall and the floor please use complete sentences and include trig

Answer 1

It depends on what the "ideal" angle is.. the angle that it's propped up against the wall.
A small angle < 45° will get more height but falling backwards off the ladder is more probable. An angle > 45° get's much less height and get's more unsafe the closer to 90° it becomes. I guess you should go with 45° as the ideal angle.

Then the ladder just makes a right triangle with the building.
Opposite side is the height, Hypotenuse is the ladder length.
Soh Cah Toa

Sin(45°) = (height) / (10 ft)
10 × Sin(45°) = height
7.07 ft