Question
57) If \( \mathrm{f}(x)=\cos 3 x+\sin x \), then \( \mathrm{f}\left(\frac{\pi}{2}\right) \) equals \( \begin{array}{llll}\text { A) } 1 & \text { B) } 2 & \text { C) }-1 & \text { D) } 0\end{array} \)
Ask by Fernandez Black. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \mathrm{f}\left(\frac{\pi}{2}\right) = 1 \), so the answer is A) 1.
Solution
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
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Simplify this solution The Deep Dive
To find \( f\left(\frac{\pi}{2}\right) \) for the function \( f(x) = \cos(3x) + \sin(x) \), we first substitute \( x = \frac{\pi}{2} \) into the function. Calculating the cosine and sine values: \[ f\left(\frac{\pi}{2}\right) = \cos\left(3 \cdot \frac{\pi}{2}\right) + \sin\left(\frac{\pi}{2}\right) \] Now, simplifying each part: 1. \( \cos\left(3 \cdot \frac{\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) = 0 \) 2. \( \sin\left(\frac{\pi}{2}\right) = 1 \) Adding these values together: \[ f\left(\frac{\pi}{2}\right) = 0 + 1 = 1 \] Thus, the result is \( \boxed{1} \).
