Find $\sin \theta$. \[ \sec \theta=\frac{6}{5}, \tan \theta<0 \] \[ \sin \theta= \] (Use integers or fractions for any numbers in the expression.)

Step 1 ：The secant of an angle is defined as the reciprocal of the cosine of the angle. So, we have \(\cos \theta = \frac{5}{6}\).

Step 2 ：The sign of the tangent of an angle depends on the quadrant in which the angle lies. Since \(\tan \theta < 0\), the angle must be in either the second or the fourth quadrant.

Step 3 ：In the second quadrant, cosine is negative and sine is positive. In the fourth quadrant, cosine is positive and sine is negative. Since we know that \(\cos \theta = \frac{5}{6}\) is positive, the angle must be in the fourth quadrant, where sine is negative.

Step 4 ：We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\).

Step 5 ：Final Answer: \(\sin \theta = \boxed{-\frac{\sqrt{11}}{6}}\)