Find tan _ if sec _ = square root of thirty seven divided by six and sin _ < 0.

Identity: sec^2(x) = 1 + tan^2(x) => tan^2(x) = sec^2 (x) - 1

sec(x) = √[37/6] => sec^2 (x) = 37/6

tan^2 (x) = 37/6 - 1 = 31/6

tan (x) = +/- √[31/6]

Given that sin (x) is negative and sec (x) is positive, we are in the fourth quadrant, so the tangent is negative, then:

tan (x) = - √[31/6] = - 2.27