Find the remaining five trigonometric functions of \( \theta \) satisfying the conditions. (Himts Draw a triangle in the appropriate quadrant. Pay attention to signs! \[ 15, \tan \theta=\frac{7}{3}, \cos \theta<0 \]

To find the remaining trigonometric functions for \( \theta \) given \( \tan \theta = \frac{7}{3} \) and \( \cos \theta < 0 \), we start by plotting the triangle. Since \( \tan \theta = \frac{7}{3} \), we can think of the triangle’s opposite side as 7 and the adjacent side as -3 (since \( \cos \theta < 0 \) indicates we're in the second quadrant). Therefore, the hypotenuse \( r \) can be calculated using the Pythagorean theorem: \[ r = \sqrt{(-3)^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}. \] Now, we can find the remaining trigonometric functions: - \( \sin \theta = \frac{7}{\sqrt{58}} \) - \( \cos \theta = \frac{-3}{\sqrt{58}} \) - \( \sec \theta = \frac{\sqrt{58}}{-3} \) - \( \csc \theta = \frac{\sqrt{58}}{7} \) - \( \cot \theta = \frac{3}{7} \) (the reciprocal of \(\tan\)) In summary: - \(\sin \theta = \frac{7}{\sqrt{58}}, \, \cos \theta = \frac{-3}{\sqrt{58}}, \, \tan \theta = \frac{7}{3}, \, \sec \theta = \frac{\sqrt{58}}{-3}, \, \csc \theta = \frac{\sqrt{58}}{7}, \, \cot \theta = \frac{3}{7}\).