57) If \( \mathrm{f}(x)=\cos 3 x+\sin x \), then \( \mathrm{f}\left(\frac{\pi}{2}\right) \) equals \( \begin{array}{llll}\text { A) } 1 & \text { B) } 2 & \text { C) }-1 & \text { D) } 0\end{array} \)

To find \( \mathrm{f}\left(\frac{\pi}{2}\right) \) for the function \( \mathrm{f}(x) = \cos(3x) + \sin(x) \), follow these steps: 1. **Substitute \( x = \frac{\pi}{2} \) into the function:** \[ \mathrm{f}\left(\frac{\pi}{2}\right) = \cos\left(3 \times \frac{\pi}{2}\right) + \sin\left(\frac{\pi}{2}\right) \] 2. **Simplify each trigonometric function:** - **Calculate \( \cos\left(\frac{3\pi}{2}\right) \):** \[ \cos\left(\frac{3\pi}{2}\right) = 0 \] (This is because \( \frac{3\pi}{2} \) corresponds to the point \((0, -1)\) on the unit circle, and cosine represents the x-coordinate.) - **Calculate \( \sin\left(\frac{\pi}{2}\right) \):** \[ \sin\left(\frac{\pi}{2}\right) = 1 \] (This is because \( \frac{\pi}{2} \) corresponds to the point \((1, 0)\) on the unit circle, and sine represents the y-coordinate.) 3. **Combine the results:** \[ \mathrm{f}\left(\frac{\pi}{2}\right) = 0 + 1 = 1 \] **Answer:** A) 1