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 use complete sentences and include trig

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