The first order of business is to see if you can get the painting through the door at all, regardless of the angle. If you can't, you saved time! I drew a doorway and labelled the dmensions as described -- after converting everything to inches for consistency. That make it 24" wide and 69" tall. Plug them into the Pythagorean Theorem and see what happens:
24² + 69² = C²
576 + 4489 = C²
5,065 = C²
√5,065 = C²
71.16" = C
This is just a tad more than needed to pass a painting 69" of the painting. So far, so good. Now, let's see what angle we need to achieve exactly that 69"...
Let's draw a right angle with the 90° at the top corner (it doesn't matter if it's left or right). The painting becomes the hypotenuse, and remains 69". The vertical side of the triangle (which will be our adjacent side) is still 67". Together, they create a Cosine. This means:
CosΘ = Adj / Hyp
= 67 / 69
Now we solve by taking the inverse Cosine of each side:
Cos-1 (CosΘ) = Cos-1(67 / 69)
Θ ≈ 13.83°
Yes, that is less than 20, so we can move that painting after all!
