Question 9, 1.4.65 HW Score: \( 53.33 \%, 8 \) o Points: 0 of 1 Find the exact value of \( \cos \theta \), given that \( \sin \theta=\frac{12}{13} \) and \( \theta \) is in quadrant II. Rationalize denominators when applicable. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \boldsymbol{\operatorname { c o s }} \theta= \) \( \square \) (Simplify your answer. Type an exact answer, using radicals as needed. Type an integer or a fraction.) B. The function is undefined.

To find the exact value of \( \cos \theta \) when \( \sin \theta = \frac{12}{13} \) and \( \theta \) is in quadrant II, we can use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Given \( \sin \theta = \frac{12}{13} \), we first calculate \( \sin^2 \theta \): \[ \sin^2 \theta = \left( \frac{12}{13} \right)^2 = \frac{144}{169}. \] Now, substitute into the Pythagorean identity: \[ \frac{144}{169} + \cos^2 \theta = 1. \] To find \( \cos^2 \theta \), we rearrange the equation: \[ \cos^2 \theta = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169}. \] Taking the square root gives us: \[ \cos \theta = \pm \frac{5}{13}. \] Since \( \theta \) is in quadrant II, where cosine is negative, we have: \[ \cos \theta = -\frac{5}{13}. \] So, the answer is: A. \( \cos \theta = -\frac{5}{13} \) --- Did you know that the sine and cosine functions originate from the study of triangles in ancient civilizations, particularly in India and Greece? These concepts were crucial in navigational techniques and astronomy, laying the foundation for modern trigonometry! Fun fact: The Pythagorean identity not only helps find cosine from sine but is also the backbone of countless real-world applications! From engineering to computer science, it helps calculate everything, from the trajectory of a rocket to the vibrations in a bridge—those triangles are everywhere!