Step 1: Identify the given information and define the sides of a right-angled triangle. Given $\text{Tan A} = \frac{3}{8}$. In a right-angled triangle, $\text{Tan A} = \frac{\text{Opposite}}{\text{Adjacent}}$. Let the Opposite side be $3x$ and the Adjacent side be $8x$ for some constant $x$. Step 2: Calculate the Hypotenuse using the Pythagorean theorem. $$ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 $$ $$ \text{Hypotenuse}^2 = (3x)^2 + (8x)^2 $$ $$ \text{Hypotenuse}^2 = 9x^2 + 64x^2 $$ $$ \text{Hypotenuse}^2 = 73x^2 $$ $$ \text{Hypotenuse} = \sqrt{73x^2} = x\sqrt{73} $$ Step 3: Calculate $\text{Sin A}$. $$ \text{Sin A} = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3x}{x\sqrt{73}} = \frac{3}{\sqrt{73}} $$ To rationalize the denominator: $$ \text{Sin A} = \frac{3}{\sqrt{73}} \times \frac{\sqrt{73}}{\sqrt{73}} = \frac{3\sqrt{73}}{73} $$ The value of $\text{Sin A}$ is $\boxed{\frac{3\sqrt{73}}{73}}$. Step 4: Calculate $\text{Cos A}$. $$ \text{Cos A} = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8x}{x\sqrt{73}} = \frac{8}{\sqrt{73}} $$ To rationalize the denominator: $$ \text{Cos A} = \frac{8}{\sqrt{73}} \times \frac{\sqrt{73}}{\sqrt{73}} = \frac{8\sqrt{73}}{73} $$ The value of $\text{Cos A}$ is $\boxed{\frac{8\sqrt{73}}{73}}$. Step 5: Calculate $\text{Sec A}$. $\text{Sec A}$ is the reciprocal of $\text{Cos A}$. $$ \text{Sec A} = \frac{1}{\text{Cos A}} = \frac{1}{\frac{8}{\sqrt{73}}} = \frac{\sqrt{73}}{8} $$ The value of $\text{Sec A}$ is $\boxed{\frac{\sqrt{73}}{8}}$. Step 6: Calculate $\text{Cosec A}$. $\text{Cosec A}$ is the reciprocal of $\text{Sin A}$. $$ \text{Cosec A} = \frac{1}{\text{Sin A}} = \frac{1}{\frac{3}{\sqrt{73}}} = \frac{\sqrt{73}}{3} $$ The value of $\text{Cosec A}$ is $\boxed{\frac{\sqrt{73}}{3}}$. Step 7: Calculate $\text{Cot A}$. $\text{Cot A}$ is the reciprocal of $\text{Tan A}$. $$ \text{Cot A} = \frac{1}{\text{Tan A}} = \frac{1}{\frac{3}{8}} = \frac{8}{3} $$ The value of $\text{Cot A}$ is $\boxed{\frac{8}{3}}$.