Find the exact value of each of the six trigonometric functions of \( \theta \), if \( (4,-2) \) 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 determine the six trigonometric functions of the angle \( \theta \) given the point \( (4, -2) \) on its terminal side, we start by calculating the radius \( r \). The radius can be found using the formula: \[ r = \sqrt{x^2 + y^2} \] Here, \( x = 4 \) and \( y = -2 \): \[ r = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \] Now we can find the values of the six trigonometric functions: 1. **Sine**: \[ \sin \theta = \frac{y}{r} = \frac{-2}{2\sqrt{5}} = \frac{-1}{\sqrt{5}} = \frac{-\sqrt{5}}{5} \quad \text{(rationalizing the denominator)} \] 2. **Cosine**: \[ \cos \theta = \frac{x}{r} = \frac{4}{2\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \quad \text{(rationalizing the denominator)} \] 3. **Tangent**: \[ \tan \theta = \frac{y}{x} = \frac{-2}{4} = -\frac{1}{2} \] 4. **Cosecant** (reciprocal of sine): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{-\sqrt{5}}{5}} = -\frac{5}{\sqrt{5}} = -\sqrt{5} \quad \text{(rationalizes to)} \] 5. **Secant** (reciprocal of cosine): \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{2\sqrt{5}}{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5}}{2} \quad \text{(rationalizes)} \] 6. **Cotangent** (reciprocal of tangent): \[ \cot \theta = \frac{1}{\tan \theta} = -2 \] Putting it all together, the exact values of the six trigonometric functions are: \[ \sin \theta = \frac{-\sqrt{5}}{5}, \quad \cos \theta = \frac{2\sqrt{5}}{5}, \quad \tan \theta = -\frac{1}{2}, \] \[ \csc \theta = -\sqrt{5}, \quad \sec \theta = \frac{\sqrt{5}}{2}, \quad \cot \theta = -2. \]