Okay, first, you have to know that the tan(A + B) = (tanA + tanB)/(1-tanAtanB). You can learn to derive this formula - or just memorize it.
Next, given an angle with a sin = 3/5, the tangent must equal 3/4. To see why, draw a right triangle and label one of the acute angles A. Since sinA=3/5 the opposite side must be 3 and the hypotenuse must be 5. The only value possible for the adjacent side is 4 since it must be a 3,4,5 right triangle (pythagorean theorem). So the tangent is opp/adj. or 3/4.
Now you have everything you need. tan(A+B)=(tanA+tanB)/(1-tanAtanB)=(3/4+radical15)/[1-(3/4)(radical15)]
(The bit about the angle being between 0 and 90 is important because without it the sin could be 3/5 but the tangent could be negative 3/4 - if the angle was in the second quadrant.)