\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
The random variable \(X\) has the density \(f(x) = \cos(x)\), on the interval \([0, \frac{\pi}{2}]\).
Find \(P(0 \leq X \leq \frac{\pi}{4})\).
\FieldText{1}{$$\begin{aligned}[t] P\left(0 \leq X \leq \frac{\pi}{4}\right) &=\int_{0}^{\frac{\pi}{4}} \cos(x) \; \mathrm d x\\&=\left[\sin(x)\right]_{0}^{\frac{\pi}{4}}\\&= \sin\left(\frac{\pi}{4}\right) - \sin(0)\\&= \frac{\sqrt{2}}{2}\\\end{aligned}$$}