Given cot A= 8/5 and that angle A is in Quadrant I, find the exact value of cos A in simplest radical form using a rational denominator.

Answer 1

Given that cot(A) = 8/5, we know that cot(A) is by definition cos(A)/sin(A) = 1/tan(A).

Therefore, cos(A) = 8/5 * sin(A) or sin(A) = 5/8 * cos(A). But how do we solve for this definition? We use the definition of sibe and cosine on the unit circle.

(cos(A)^2 + sin(A)^2) = 1

With our relationship, we have cos(A)^2 = 64/25 * sin(A)^2. Plugging into the unit circle definition for our relationship of sin(A) gives...

cos(A)^2 * 25/64 + cos(A)^2 = 1

cos(A)^2 * (1 + 25/64) = 1

cos(A)^2 = 1/(1 + 25/64) = 1/(89/64) = 64/89 = 64/(25 + 64)

Therefore, since cos(A) is in the first quadrant, we choose the positive sqrt.

cos(A) = 8/sqrt(89) = 8/89 * sqrt(89).