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 \( \theta \).

To find the exact values of the six trigonometric functions of angle \( \theta \), given that the point \( (-4, -5) \) lies on the terminal side of the angle, we begin by determining the radius \( r \): \[ r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41} \] Now we can use the coordinates \( (x, y) = (-4, -5) \) along with the radius \( r = \sqrt{41} \) to find the trigonometric functions. 1. **Sine:** \[ \sin(\theta) = \frac{y}{r} = \frac{-5}{\sqrt{41}} \] 2. **Cosine:** \[ \cos(\theta) = \frac{x}{r} = \frac{-4}{\sqrt{41}} \] 3. **Tangent:** \[ \tan(\theta) = \frac{y}{x} = \frac{-5}{-4} = \frac{5}{4} \] 4. **Cosecant (the reciprocal of sine):** \[ \csc(\theta) = \frac{r}{y} = \frac{\sqrt{41}}{-5} = -\frac{\sqrt{41}}{5} \] 5. **Secant (the reciprocal of cosine):** \[ \sec(\theta) = \frac{r}{x} = \frac{\sqrt{41}}{-4} = -\frac{\sqrt{41}}{4} \] 6. **Cotangent (the reciprocal of tangent):** \[ \cot(\theta) = \frac{x}{y} = \frac{-4}{-5} = \frac{4}{5} \] Summarizing the values of the six trigonometric functions: \[ \sin(\theta) = -\frac{5}{\sqrt{41}}, \quad \cos(\theta) = -\frac{4}{\sqrt{41}}, \quad \tan(\theta) = \frac{5}{4} \] \[ \csc(\theta) = -\frac{\sqrt{41}}{5}, \quad \sec(\theta) = -\frac{\sqrt{41}}{4}, \quad \cot(\theta) = \frac{4}{5} \]