Find the exact value of each of the six trigonometric functions of \( \theta \), if \( (-2,-7) \) 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 denominator

To find the exact values of the six trigonometric functions of angle \( \theta \) when the point \((-2, -7)\) lies on the terminal side of angle \( \theta \), we can use the coordinates of the point to determine the values of the trigonometric functions. Given that the point \((-2, -7)\) lies on the terminal side of angle \( \theta \), we can determine the values of the trigonometric functions as follows: 1. \( \sin \theta = \frac{y}{r} \) 2. \( \cos \theta = \frac{x}{r} \) 3. \( \tan \theta = \frac{y}{x} \) 4. \( \csc \theta = \frac{r}{y} \) 5. \( \sec \theta = \frac{r}{x} \) 6. \( \cot \theta = \frac{x}{y} \) where \( x = -2 \), \( y = -7 \), and \( r \) is the distance from the origin to the point \((-2, -7)\), which can be calculated using the Pythagorean theorem. Let's calculate the values of the trigonometric functions step by step: 1. Calculate \( r \): \[ r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53} \] 2. Calculate \( \sin \theta \): \[ \sin \theta = \frac{y}{r} = \frac{-7}{\sqrt{53}} \] 3. Calculate \( \cos \theta \): \[ \cos \theta = \frac{x}{r} = \frac{-2}{\sqrt{53}} \] 4. Calculate \( \tan \theta \): \[ \tan \theta = \frac{y}{x} = \frac{-7}{-2} = \frac{7}{2} \] 5. Calculate \( \csc \theta \): \[ \csc \theta = \frac{r}{y} = \frac{\sqrt{53}}{-7} = -\frac{\sqrt{53}}{7} \] 6. Calculate \( \sec \theta \): \[ \sec \theta = \frac{r}{x} = \frac{\sqrt{53}}{-2} = -\frac{\sqrt{53}}{2} \] 7. Calculate \( \cot \theta \): \[ \cot \theta = \frac{x}{y} = \frac{-2}{-7} = \frac{2}{7} \] Therefore, the exact values of the six trigonometric functions of angle \( \theta \) are: - \( \sin \theta = \frac{-7}{\sqrt{53}} \) - \( \cos \theta = \frac{-2}{\sqrt{53}} \) - \( \tan \theta = \frac{7}{2} \) - \( \csc \theta = -\frac{\sqrt{53}}{7} \) - \( \sec \theta = -\frac{\sqrt{53}}{2} \) - \( \cot \theta = \frac{2}{7} \)