Find the maximum value of \(2 + \sin (\theta + 25)\).
Answer Details
The value of \(\sin(\theta + 25)\) varies between -1 and 1. Therefore, the maximum value of \(2 + \sin(\theta + 25)\) occurs when \(\sin(\theta + 25) = 1\). This occurs when \(\theta + 25 = \frac{\pi}{2} + 2n\pi\), where \(n\) is an integer. Thus, \(\theta = \frac{\pi}{2} - 25 + 2n\pi\). Substituting this value of \(\theta\) into the expression for \(2 + \sin(\theta + 25)\), we get:
\[2 + \sin\left(\frac{\pi}{2} + 2n\pi\right) = 2 + 1 = 3\]
Therefore, the maximum value of \(2 + \sin(\theta + 25)\) is 3.