Step 1 :We are given that \(\sin \theta = -\frac{12}{13}\) and \(\tan \theta < 0\).
Step 2 :Since the sine is negative and the tangent is negative, this means the angle is in the third quadrant. In the third quadrant, cosine is also negative.
Step 3 :We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find the cosine of the angle.
Step 4 :Substituting the given value of sine into the Pythagorean identity, we get \((-\frac{12}{13})^2 + \cos^2 \theta = 1\).
Step 5 :Solving for \(\cos \theta\), we get \(\cos \theta = -\frac{5}{13}\).
Step 6 :The secant of an angle is the reciprocal of the cosine of the angle. So, \(\sec \theta = -\frac{1}{\cos \theta}\).
Step 7 :Substituting the value of \(\cos \theta\) into the equation, we get \(\sec \theta = -\frac{1}{-\frac{5}{13}}\).
Step 8 :Solving for \(\sec \theta\), we get \(\sec \theta = -2.6\).
Step 9 :So, the final answer is \(\boxed{-2.6}\).