If \( \tan \theta=\frac{12}{5} \) and \( 90^{\circ}

Ask by Cross Joseph. in South Africa

Mar 14,2025

Given \( \tan \theta = \frac{12}{5} \), we can establish a right triangle where the opposite side is \( 12 \) and the adjacent side is \( 5 \). We can find the hypotenuse \( r \) using the Pythagorean theorem: \[ r = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] Now, we can find \( \sin \theta \) and \( \cos \theta \): \[ \sin \theta = \frac{12}{13}, \quad \cos \theta = \frac{5}{13} \] Now, since \( 90^{\circ} < \theta < 270^{\circ} \), this implies that \( \theta \) is in the second or third quadrant where cosine is negative. Thus, we write: \[ \cos \theta = -\frac{5}{13} \] Next, we need to calculate \( \frac{10}{\cos \theta} - \frac{5}{\cos \theta} \). We first note that: \[ \frac{10}{\cos \theta} = \frac{10}{-\frac{5}{13}} = 10 \cdot \left(-\frac{13}{5}\right) = -\frac{130}{5} = -26 \] and \[ \frac{5}{\cos \theta} = \frac{5}{-\frac{5}{13}} = 5 \cdot \left(-\frac{13}{5}\right) = -13 \] Now, we calculate: \[ \frac{10}{\cos \theta} - \frac{5}{\cos \theta} = -26 - (-13) = -26 + 13 = -13 \] Thus, the final result is: \[ \boxed{-13} \]