Given that tangent, A, equals, 3tanA=3 and tangent, B, equals, 1tanB=1, and that angles AA and BB are both in Quadrant I, find the exact value of sine, left bracket, A, minus, B, right bracketsin(A−B), in simplest radical form.

The problem is asking for the exact value of sin(A-B) given that tanA=3 and tanB=1, with both angles A and B in Quadrant I.

Step 1: We know that tanA = sinA/cosA and tanB = sinB/cosB. Since tanA = 3, this implies that sinA = 3cosA. Similarly, since tanB = 1, this implies that sinB = cosB.

Step 2: We also know that sin^2A + cos^2A = 1 and sin^2B + cos^2B = 1. Substituting sinA = 3cosA and sinB = cosB into these equations, we get (3cosA)^2 + cos^2A = 1 and (cosB)^2 + cos^2B = 1. Solving these equations, we find that cosA = sqrt(1/10) and cosB = sqrt(1/2).

Step 3: Now we can find sinA and sinB by substituting cosA and cosB into the equations sinA = 3cosA and sinB = cosB. We find that sinA = 3sqrt(1/10) = sqrt(9/10) and sinB = sqrt(1/2).

Step 4: Finally, we can find sin(A-B) using the identity sin(A-B) = sinAcosB - cosAsinB. Substituting the values we found for sinA, sinB, cosA, and cosB, we get sin(A-B) = sqrt(9/10)*sqrt(1/2) - sqrt(1/10)*sqrt(1/2) = sqrt(9/20) - sqrt(1/20) = sqrt(8/20) = sqrt(2/5).

So, the exact value of sin(A-B) in simplest radical form is sqrt(2/5).

The problem is asking for the exact value of sin(A-B) given that tanA=3 and tanB=1, with both angles A and B in Quadrant I.

Step 1: We know that tanA = sinA/cosA and tanB = sinB/cosB.

Step 2: Since tanA = 3, we can form a right triangle with opposite side 3, adjacent side 1, and hypotenuse sqrt(1^2 + 3^2) = sqrt(10). So, sinA = 3/sqrt(10) = sqrt(10)/10 and cosA = 1/sqrt(10) = sqrt(10)/10.

Step 3: Similarly, since tanB = 1, we can form a right triangle with opposite side 1, adjacent side 1, and hypotenuse sqrt(1^2 + 1^2) = sqrt(2). So, sinB = 1/sqrt(2) = sqrt(2)/2 and cosB = 1/sqrt(2) = sqrt(2)/2.

Step 4: Now we can use the formula for sin(A-B) = sinAcosB - cosAsinB.

Step 5: Substituting the values we found, sin(A-B) = (sqrt(10)/10)(sqrt(2)/2) - (sqrt(10)/10)(sqrt(2)/2) = 0.

So, the exact value of sin(A-B) is 0.