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Given cot A = 21/20 and that angle A is in Quadrant I, find the exact value of A in simplest radical form using a rational denominator.

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User ThisGuy
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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)

User Bhagyas
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