1. The problem is to find the value of $\sinh\left(3+\frac{\pi i}{6}\right)$. 2. Recall the formula for the hyperbolic sine of a complex number $z = x + yi$: $$\sinh(x+yi) = \sinh(x)\cos(y) + i\cosh(x)\sin(y)$$ where $x$ and $y$ are real numbers. 3. Here, $x=3$ and $y=\frac{\pi}{6}$. 4. Calculate each part: - $\sinh(3)$ (hyperbolic sine of 3) - $\cos\left(\frac{\pi}{6}\right)$ (cosine of $\pi/6$) - $\cosh(3)$ (hyperbolic cosine of 3) - $\sin\left(\frac{\pi}{6}\right)$ (sine of $\pi/6$) 5. Using known values: - $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ - $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ 6. Substitute and write the expression: $$\sinh\left(3+\frac{\pi i}{6}\right) = \sinh(3) \cdot \frac{\sqrt{3}}{2} + i \cdot \cosh(3) \cdot \frac{1}{2}$$ 7. This is the exact form. For numerical approximation: - $\sinh(3) \approx 10.0179$ - $\cosh(3) \approx 10.0677$ 8. Therefore: $$\sinh\left(3+\frac{\pi i}{6}\right) \approx 10.0179 \times \frac{\sqrt{3}}{2} + i \times 10.0677 \times \frac{1}{2}$$ 9. Calculate: - Real part: $10.0179 \times 0.8660 \approx 8.674$ - Imaginary part: $10.0677 \times 0.5 \approx 5.034$ 10. Final answer: $$\sinh\left(3+\frac{\pi i}{6}\right) \approx 8.674 + 5.034i$$