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

Ask by Franklin Gibbs. in South Africa

Mar 06,2025

Let's solve the problem step by step. ### Step 1. Simplify the Expression We are given the expression \[ \frac{10}{\cos \theta} - \frac{5}{\cos \theta}. \] Since the denominators are the same, we can combine the numerators: \[ \frac{10}{\cos \theta} - \frac{5}{\cos \theta} = \frac{10 - 5}{\cos \theta} = \frac{5}{\cos \theta}. \] ### Step 2. Determine \(\cos \theta\) from \(\tan \theta\) We are given \[ \tan \theta = \frac{12}{5}. \] A triangle corresponding to this relation has: - Opposite side \(= 12\) - Adjacent side \(= 5\) The hypotenuse of the triangle is \[ \text{hypotenuse} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13. \] Thus, if \(\theta\) were in the first quadrant, \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13}. \] However, we are given that \(90^\circ < \theta < 270^\circ\). In this interval, the angle is either in the second or third quadrant. Since \(\tan \theta = \frac{12}{5}\) is positive, \(\theta\) must be in the **third quadrant** (because tangent is positive in the first and third quadrants, and the first quadrant is excluded by the interval). In the third quadrant, the cosine is negative. Therefore, \[ \cos \theta = -\frac{5}{13}. \] ### Step 3. Substitute \(\cos \theta\) into the Expression Now substitute the value of \(\cos \theta\) into the simplified expression: \[ \frac{5}{\cos \theta} = \frac{5}{-\frac{5}{13}}. \] Simplify the fraction: \[ \frac{5}{-\frac{5}{13}} = 5 \times \left(-\frac{13}{5}\right) = -13. \] ### Final Answer The value of the given expression is: \[ \boxed{-13}. \]