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 find the six trigonometric functions of the angle \( \theta \) given the point \( (4, -2) \) on the terminal side, we first need to determine the radius \( r \) using the coordinates \( (x, y) = (4, -2) \). The radius \( r \) is calculated using the formula: \[ r = \sqrt{x^2 + y^2} \] Substituting the values of \( x \) and \( y \): \[ r = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \] Now we can find 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** (the reciprocal of sine): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{-\sqrt{5}}{5}} = -\frac{5}{\sqrt{5}} = -\sqrt{5} \quad \text{(rationalizing the denominator)} \] 5. **Secant** (the reciprocal of cosine): \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{2\sqrt{5}}{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5}}{2} \quad \text{(rationalizing the denominator)} \] 6. **Cotangent** (the reciprocal of tangent): \[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{1}{2}} = -2 \] Now we summarize the results: - \( \sin \theta = \frac{-\sqrt{5}}{5} \) - \( \cos \theta = \frac{2\sqrt{5}}{5} \) - \( \tan \theta = -\frac{1}{2} \) - \( \csc \theta = -\sqrt{5} \) - \( \sec \theta = \frac{\sqrt{5}}{2} \) - \( \cot \theta = -2 \) Thus, the exact value of \( \sin \theta \) is: \[ \sin \theta = \frac{-\sqrt{5}}{5} \]