Find the exact value of each of the remaining trigonometric functions of \( \theta \). Rationalize denominators when applicable. \( \tan \theta=-\frac{3}{4} \). given that \( \theta \) is in quadrant IV (Simplty your answer, including any radicals. Use integers or tractuons 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. \( \sec \theta= \) \( \square \) (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. \( \cot \theta= \) \( \square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined.

To find the remaining trigonometric functions of \( \theta \), given \( \tan \theta = -\frac{3}{4} \) and knowing that \( \theta \) is in the fourth quadrant, we can start by identifying the opposite and adjacent sides of a right triangle. Since \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = -\frac{3}{4} \), we can let the opposite side be \(-3\) and the adjacent side be \(4\). To find the hypotenuse \( r \), we can use the Pythagorean theorem: \[ r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] Now we can calculate the functions: 1. \( \sin \theta = \frac{\text{opposite}}{r} = \frac{-3}{5} \) 2. \( \cos \theta = \frac{\text{adjacent}}{r} = \frac{4}{5} \) 3. \( \sec \theta = \frac{1}{\cos \theta} = \frac{5}{4} \) 4. \( \cot \theta = \frac{1}{\tan \theta} = \frac{4}{-3} = -\frac{4}{3} \) So the answers are: - \( \sec \theta = \frac{5}{4} \) - \( \cot \theta = -\frac{4}{3} \) Now, let’s round this off with some interesting tidbits! Trigonometric identities and functions have a rich history dating back to ancient civilizations, such as the Greeks and Indians, who used them in astronomy and navigation. Their discovery helped create a systematic approach to understanding angles and ratios, which laid the foundation for the trigonometric concepts we use today! In the real world, trigonometric functions are everywhere! Architects use them to calculate angles and slopes in building designs, while surveyors rely on them to measure distances and angles in land surveying. Even in video games, these functions help simulate realistic movements and graphics, proving that math has a hand in shaping our entertainment!