Answer:
We can start by using the Pythagorean identity to simplify the expression for cos J:
cos^2(J) + sin^2(J) = 1
Since we are given the value of sin J, we can substitute and solve for cos J:
cos^2(J) + (4/√82)^2 = 1
cos^2(J) + 16/82 = 1
cos^2(J) = 66/82
cos(J) = ±√(66/82)
We want to express cos J in simplest radical form, so we can simplify the square root by factoring out the greatest perfect square factor of the numerator:
cos(J) = ±√[(2311)/(2*41)]
cos(J) = ±(√2/2) * (√33/√41)
Since J is in the first or second quadrant (based on the given value of sin J), we know that cos J is positive, so we can drop the negative sign:
cos(J) = (√2/2) * (√33/√41)
Therefore, the exact value of cos J in simplest radical form is (√2/2) * (√33/√41).