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

Ask by Franklin Gibbs. in South Africa

Mar 06,2025

Let's start with the given information: \( \tan \theta = \frac{12}{5} \). We need to find \( \frac{10}{\cos \theta} - \frac{5}{\cos \theta} \). First, we can simplify the expression: \[ \frac{10}{\cos \theta} - \frac{5}{\cos \theta} = \frac{10 - 5}{\cos \theta} = \frac{5}{\cos \theta} \] Next, we need to determine the cosine of \( \theta \). Since \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5} \), we can represent the opposite and adjacent sides of a right triangle. The opposite side is 12, and the adjacent side is 5. Using the Pythagorean theorem: \[ \text{hypotenuse} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] Now we can find \( \cos \theta \): \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13} \] Now we substitute this value into our earlier expression: \[ \frac{5}{\cos \theta} = \frac{5}{\frac{5}{13}} = 5 \times \frac{13}{5} = 13 \] Thus, the value of \( \frac{10}{\cos \theta} - \frac{5}{\cos \theta} \) is \( \boxed{13} \).