Answer:
We know that cot A = adjacent side / opposite side = 21/20.
Let x be the length of the adjacent side and y be the length of the opposite side.
Using the Pythagorean theorem, we have:
x^2 + y^2 = (20k)^2, where k is some positive constant.
Since angle A is in Quadrant I, x is positive and y is positive. Thus, we can take the square root of both sides to get:
x + y = 20k.
Now, we can use the fact that cot A = adjacent side / opposite side = 21/20 to get:
x/y = 21/20.
Multiplying both sides by y, we get:
x = (21/20)y.
Substituting this into x + y = 20k, we get:
(21/20)y + y = 20k
Simplifying, we get:
(41/20)y = 20k
y = (20k)/(41/20) = (400k)/41
Now, we can use the Pythagorean theorem to find x:
x^2 + y^2 = (20k)^2
(21/20)y^2 + y^2 = (20k)^2
(441/400)y^2 = (20k)^2
y^2 = (400/441)(20k)^2
y = (20k/441)sqrt(441 x 20)
y = (20k/21)sqrt(20)
Therefore, angle A in simplest radical form using a rational denominator is given by:
tan A = y/x = [(20k/21)sqrt(20)] / [(21/20)y] = sqrt(20)/21
A = arctan(sqrt(20)/21)