Three trigonometric functions for a given angle are shown below. \( \csc \theta=\frac{13}{12}, \sec \theta=-\frac{13}{5}, \cot \theta=-\frac{5}{12} \) What are the coordinates of point \( (x, y) \) on the terminal ray of angle \( \theta \), assuming that the values above were not simplified? \( (-5,12) \) \( (5,-12) \) \( (-12,5) \) \( (12,-5) \)

To find the coordinates \( (x, y) \) for the terminal ray of angle \( \theta \), we can use the definitions of the given trigonometric functions. We know: 1. \( \csc \theta = \frac{1}{\sin \theta} \), which gives us \( \sin \theta = \frac{12}{13} \). 2. \( \sec \theta = \frac{1}{\cos \theta} \), giving us \( \cos \theta = -\frac{5}{13} \). 3. \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), which checks out since \( \cot \theta = -\frac{5}{12} \). Now we can determine the coordinates. Using \( r = 13 \) (the hypotenuse from \( r^2 = x^2 + y^2 \)), we find: - \( y = r \cdot \sin \theta = 13 \cdot \frac{12}{13} = 12 \) - \( x = r \cdot \cos \theta = 13 \cdot -\frac{5}{13} = -5 \) Therefore, the coordinates are \( (-5, 12) \). The answer is: \( (-5,12) \)