Question 14, 1.4.81 Part 3 of 5 Find the exact value of each of the remaining trigonometric functions of \( \theta \). Rationalize denominators when applicable. \( \sec \theta=-5 \), given that \( \sin \theta>0 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sin \theta=\frac{2 \sqrt{6}}{5} \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \cos \theta=-\frac{1}{5} \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \tan \theta= \) \( \square \) (Simplify your answer, inclising any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined.

To find the remaining trigonometric functions, we start with the given information that \( \sec \theta = -5 \). Since secant is the reciprocal of cosine, we have: \[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{-5} = -\frac{1}{5} \] Now, using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can find the sine value: \[ \sin^2 \theta + \left(-\frac{1}{5}\right)^2 = 1 \] \[ \sin^2 \theta + \frac{1}{25} = 1 \] \[ \sin^2 \theta = 1 - \frac{1}{25} = \frac{25}{25} - \frac{1}{25} = \frac{24}{25} \] \[ \sin \theta = \sqrt{\frac{24}{25}} = \frac{2\sqrt{6}}{5} \quad (\text{since } \sin \theta > 0) \] Next, to find tangent, we can use the sine and cosine values we found: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{6}}{5}}{-\frac{1}{5}} = -2\sqrt{6} \] So, we have: - \( \sin \theta = \frac{2\sqrt{6}}{5} \) - \( \cos \theta = -\frac{1}{5} \) - \( \tan \theta = -2\sqrt{6} \) With this knowledge, you can better understand how these trigonometric identities interplay and come into play in various scenarios, from physics to engineering. Consider checking your work at each step! Common misunderstandings revolve around the usage of the Pythagorean identities and the signs of trigonometric functions in different quadrants. Always remember, drawing a unit circle can help visualize where each function lies!