1. **State the problem:** Find the approximate value of $$\cot(\csc^{-1}(3.6))$$. 2. **Recall definitions and formulas:** - The inverse cosecant function $$\csc^{-1}(x)$$ gives an angle $$\theta$$ such that $$\csc(\theta) = x$$. - Since $$\csc(\theta) = \frac{1}{\sin(\theta)}$$, we have $$\sin(\theta) = \frac{1}{x}$$. - The cotangent function is $$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$$. 3. **Find $$\theta$$:** Given $$\csc^{-1}(3.6) = \theta$$, so $$\sin(\theta) = \frac{1}{3.6} \approx 0.2777777$$. 4. **Find $$\cos(\theta)$$ using Pythagorean identity:** $$\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{1}{3.6}\right)^2} = \sqrt{1 - \frac{1}{12.96}} = \sqrt{\frac{12.96 - 1}{12.96}} = \sqrt{\frac{11.96}{12.96}}$$ 5. **Simplify:** $$\cos(\theta) = \sqrt{\frac{11.96}{12.96}} = \frac{\sqrt{11.96}}{\sqrt{12.96}}$$ 6. **Calculate approximate values:** $$\sqrt{11.96} \approx 3.458$$ $$\sqrt{12.96} = 3.6$$ 7. **Calculate $$\cot(\theta)$$:** $$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{3.458/3.6}{1/3.6} = \frac{3.458}{3.6} \times 3.6 = 3.458$$ 8. **Final answer rounded to four decimal places:** $$\cot(\csc^{-1}(3.6)) \approx 3.4583$$