Find the exact value of each of the six trigonometric functions of \( \theta \), if \( (-4,-5) \) is a point on the terminal side of angle \( \theta \). \( \sin \theta=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

To find the trigonometric functions of \( \theta \), we first determine the radius \( r \) using the coordinates of the point \( (-4, -5) \). We can calculate \( r \) with the formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}. \] Now we can find the six trigonometric functions: 1. **Sine**: \[ \sin \theta = \frac{y}{r} = \frac{-5}{\sqrt{41}} = -\frac{5}{\sqrt{41}}. \] Rationalizing the denominator, we get: \[ \sin \theta = -\frac{5 \sqrt{41}}{41}. \] 2. **Cosine**: \[ \cos \theta = \frac{x}{r} = \frac{-4}{\sqrt{41}} = -\frac{4}{\sqrt{41}}. \] Rationalizing the denominator gives: \[ \cos \theta = -\frac{4 \sqrt{41}}{41}. \] 3. **Tangent**: \[ \tan \theta = \frac{y}{x} = \frac{-5}{-4} = \frac{5}{4}. \] 4. **Cosecant**: \[ \csc \theta = \frac{1}{\sin \theta} = -\frac{\sqrt{41}}{5}. \] 5. **Secant**: \[ \sec \theta = \frac{1}{\cos \theta} = -\frac{\sqrt{41}}{4}. \] 6. **Cotangent**: \[ \cot \theta = \frac{1}{\tan \theta} = \frac{4}{5}. \] Thus, the exact values of the six trigonometric functions are: - \( \sin \theta = -\frac{5 \sqrt{41}}{41} \), - \( \cos \theta = -\frac{4 \sqrt{41}}{41} \), - \( \tan \theta = \frac{5}{4} \), - \( \csc \theta = -\frac{\sqrt{41}}{5} \), - \( \sec \theta = -\frac{\sqrt{41}}{4} \), - \( \cot \theta = \frac{4}{5} \).