To find the length of the third leg of the triangle use pythageroean’s therom: a^2 + b^2 = c^2 with a and b being the legs, and c being the hypotenuse. Useing algebra, substite the values you know from sinx = 3/4 and you get:

a^2 + 3^2 = 4^2

a^2 + 9 = 16

a^2 = 7

a = 7^(1/2) or the square root of 7.

The second part to this problem is cosx<0. When is a number less than 0? When it's a negative number. When is cosx negative? In the second and third quadrants. Quad 1: All - All trig functions are positive Quad 2: "S"tudents - Only Sin is positive. Quad 3: "T"ake - Only tan is positive. Quad 4: "C"alculus. Only cos is positive. Since cosx is negative it must be in the second or third quadrants. But, we know sin is positive (as given in the problem) and Sin is only positive in the first and second quadrants. This means that this takes place in the second quadrant. Tanx is the ratio of the opposite and adjacent (opp over adj) legs in the second quadrant. So tanx= -3/sq. root 7 or -1.13 You add a negative because tangent isn't positive in the second quadrant. I hope this wasn't too confusing!