Find the exact value of each of the remaining trigonometric functions of \( \theta \). Rationalize denominators when applicable. \( \tan \theta=\frac{4}{3} \), given that \( \theta \) is in quadrant I Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \boldsymbol{\operatorname { s i n }} \theta=\frac{4}{5} \) (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. \( \cos \theta= \) \( \square \) (Simplify your answer, including any radicals. Use integers or fractiofs for any numbers in the expression.) B. The function is undefined.
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To find the remaining trigonometric functions given \( \tan \theta = \frac{4}{3} \), we can use the definitions of sine and cosine in relation to tangent. Since we know that \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \), we can represent the opposite side as 4 and the adjacent side as 3. Using the Pythagorean theorem, the hypotenuse \( r \) can be calculated as follows: \[ r = \sqrt{\text{opposite}^2 + \text{adjacent}^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] Now, we can find \( \sin \theta \) and \( \cos \theta \): \[ \sin \theta = \frac{\text{opposite}}{r} = \frac{4}{5} \] \[ \cos \theta = \frac{\text{adjacent}}{r} = \frac{3}{5} \] So the answers are: For \( \sin \theta \): A. \( \boldsymbol{\operatorname { s i n }} \theta=\frac{4}{5} \) For \( \cos \theta \): A. \( \cos \theta= \frac{3}{5} \)
