ACT Math: Given sin θ = -5/13 in Quadrant III, Find tan θ + cos θ

Exact Question Text

Given that $\sin\theta = -\frac{5}{13}$ and $\pi < \theta < \frac{3\pi}{2}$ (Quadrant III), what is the value of $\tan\theta + \cos\theta$?

Step-by-Step Solution

1. Pythagorean Identity:

$$ \cos^2\theta = 1 - \sin^2\theta = 1 - \left(-\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} $$ In Quadrant III, cosine is negative $\implies \cos\theta = -\frac{12}{13}$.

1. Compute $\tan\theta$:

$$ \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{-5/13}{-12/13} = +\frac{5}{12} $$

1. Evaluate the Expression:

$$ \tan\theta + \cos\theta = \frac{5}{12} + \left(-\frac{12}{13}\right) = \frac{65 - 144}{156} = -\mathbf{\frac{79}{156}} $$