1. Stating the problem: We need to find the reciprocal of the mixed number $3 \frac{3}{4}$.\n\n2. Converting the mixed number to an improper fraction: \n$3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$.\n\n3. Understanding the reciprocal: \nThe reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.\n\n4. Finding the reciprocal: \nReciprocal of $\frac{15}{4}$ is $\frac{4}{15}$.\n\n5. Final answer: \nThe reciprocal of $3 \frac{3}{4}$ is $\frac{4}{15}$.